LMFDB - Embedded newform 8624.2.a.df.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8624,2,Mod(1,8624)] 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("8624.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,0,0,0,0,22,0,-10,0,0,0,-8,0,0,0,0,0,0,0,-4,0,18,0,0, 0,12,0,0,0,0,0,0,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

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

Self dual:	yes
Analytic conductor:	$68.8629867032$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 26x^{8} + 245x^{6} - 1038x^{4} + 1884x^{2} - 968 $ x^10 - 26x^8 + 245x^6 - 1038x^4 + 1884x^2 - 968
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 539)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.15293$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.15293 q^{3} -3.87589 q^{5} +1.63513 q^{9} -1.00000 q^{11} +4.09860 q^{13} +8.34453 q^{15} -0.824752 q^{17} -4.50196 q^{19} -4.86320 q^{23} +10.0225 q^{25} +2.93849 q^{27} +7.79629 q^{29} -3.82646 q^{31} +2.15293 q^{33} +8.11646 q^{37} -8.82401 q^{39} -11.2329 q^{41} +1.86134 q^{43} -6.33756 q^{45} -7.69973 q^{47} +1.77564 q^{51} -3.93495 q^{53} +3.87589 q^{55} +9.69242 q^{57} +9.45193 q^{59} -8.40447 q^{61} -15.8857 q^{65} -9.45914 q^{67} +10.4701 q^{69} -13.4591 q^{71} +6.19352 q^{73} -21.5778 q^{75} -0.868405 q^{79} -11.2317 q^{81} -8.19720 q^{83} +3.19665 q^{85} -16.7849 q^{87} -3.87589 q^{89} +8.23813 q^{93} +17.4491 q^{95} -2.10351 q^{97} -1.63513 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 22 q^{9} - 10 q^{11} - 8 q^{15} - 4 q^{23} + 18 q^{25} + 12 q^{29} + 40 q^{37} + 16 q^{39} + 8 q^{43} + 16 q^{53} - 8 q^{57} - 32 q^{65} + 4 q^{67} - 36 q^{71} - 8 q^{79} - 6 q^{81} + 88 q^{85} + 44 q^{93}+ \cdots - 22 q^{99}+O(q^{100}) $ 10 * q + 22 * q^9 - 10 * q^11 - 8 * q^15 - 4 * q^23 + 18 * q^25 + 12 * q^29 + 40 * q^37 + 16 * q^39 + 8 * q^43 + 16 * q^53 - 8 * q^57 - 32 * q^65 + 4 * q^67 - 36 * q^71 - 8 * q^79 - 6 * q^81 + 88 * q^85 + 44 * q^93 + 64 * q^95 - 22 * 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
$3$	−2.15293	−1.24300	−0.621499	−	0.783415i	$-0.713476\pi$
$3$	−2.15293	−1.24300	−0.621499	+	0.783415i	$0.713476\pi$
$4$	0	0
$5$	−3.87589	−1.73335	−0.866675	−	0.498873i	$-0.833747\pi$
$5$	−3.87589	−1.73335	−0.866675	+	0.498873i	$0.833747\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.63513	0.545042
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.09860	1.13675	0.568373	−	0.822771i	$-0.307573\pi$
$13$	4.09860	1.13675	0.568373	+	0.822771i	$0.307573\pi$
$14$	0	0
$15$	8.34453	2.15455
$16$	0	0
$17$	−0.824752	−0.200032	−0.100016	−	0.994986i	$-0.531889\pi$
$17$	−0.824752	−0.200032	−0.100016	+	0.994986i	$0.531889\pi$
$18$	0	0
$19$	−4.50196	−1.03282	−0.516410	−	0.856342i	$-0.672732\pi$
$19$	−4.50196	−1.03282	−0.516410	+	0.856342i	$0.672732\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.86320	−1.01405	−0.507023	−	0.861932i	$-0.669254\pi$
$23$	−4.86320	−1.01405	−0.507023	+	0.861932i	$0.669254\pi$
$24$	0	0
$25$	10.0225	2.00450
$26$	0	0
$27$	2.93849	0.565512
$28$	0	0
$29$	7.79629	1.44774	0.723868	−	0.689939i	$-0.242363\pi$
$29$	7.79629	1.44774	0.723868	+	0.689939i	$0.242363\pi$
$30$	0	0
$31$	−3.82646	−0.687253	−0.343627	−	0.939106i	$-0.611655\pi$
$31$	−3.82646	−0.687253	−0.343627	+	0.939106i	$0.611655\pi$
$32$	0	0
$33$	2.15293	0.374778
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.11646	1.33434	0.667169	−	0.744907i	$-0.267506\pi$
$37$	8.11646	1.33434	0.667169	+	0.744907i	$0.267506\pi$
$38$	0	0
$39$	−8.82401	−1.41297
$40$	0	0
$41$	−11.2329	−1.75428	−0.877142	−	0.480232i	$-0.840553\pi$
$41$	−11.2329	−1.75428	−0.877142	+	0.480232i	$0.840553\pi$
$42$	0	0
$43$	1.86134	0.283852	0.141926	−	0.989877i	$-0.454670\pi$
$43$	1.86134	0.283852	0.141926	+	0.989877i	$0.454670\pi$
$44$	0	0
$45$	−6.33756	−0.944748
$46$	0	0
$47$	−7.69973	−1.12312	−0.561561	−	0.827436i	$-0.689799\pi$
$47$	−7.69973	−1.12312	−0.561561	+	0.827436i	$0.689799\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.77564	0.248639
$52$	0	0
$53$	−3.93495	−0.540507	−0.270253	−	0.962789i	$-0.587107\pi$
$53$	−3.93495	−0.540507	−0.270253	+	0.962789i	$0.587107\pi$
$54$	0	0
$55$	3.87589	0.522625
$56$	0	0
$57$	9.69242	1.28379
$58$	0	0
$59$	9.45193	1.23054	0.615268	−	0.788318i	$-0.289048\pi$
$59$	9.45193	1.23054	0.615268	+	0.788318i	$0.289048\pi$
$60$	0	0
$61$	−8.40447	−1.07608	−0.538041	−	0.842919i	$-0.680835\pi$
$61$	−8.40447	−1.07608	−0.538041	+	0.842919i	$0.680835\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−15.8857	−1.97038
$66$	0	0
$67$	−9.45914	−1.15562	−0.577809	−	0.816172i	$-0.696092\pi$
$67$	−9.45914	−1.15562	−0.577809	+	0.816172i	$0.696092\pi$
$68$	0	0
$69$	10.4701	1.26046
$70$	0	0
$71$	−13.4591	−1.59731	−0.798653	−	0.601792i	$-0.794454\pi$
$71$	−13.4591	−1.59731	−0.798653	+	0.601792i	$0.794454\pi$
$72$	0	0
$73$	6.19352	0.724897	0.362448	−	0.932004i	$-0.381941\pi$
$73$	6.19352	0.724897	0.362448	+	0.932004i	$0.381941\pi$
$74$	0	0
$75$	−21.5778	−2.49159
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.868405	−0.0977032	−0.0488516	−	0.998806i	$-0.515556\pi$
$79$	−0.868405	−0.0977032	−0.0488516	+	0.998806i	$0.515556\pi$
$80$	0	0
$81$	−11.2317	−1.24797
$82$	0	0
$83$	−8.19720	−0.899759	−0.449880	−	0.893089i	$-0.648533\pi$
$83$	−8.19720	−0.899759	−0.449880	+	0.893089i	$0.648533\pi$
$84$	0	0
$85$	3.19665	0.346725
$86$	0	0
$87$	−16.7849	−1.79953
$88$	0	0
$89$	−3.87589	−0.410843	−0.205422	−	0.978674i	$-0.565857\pi$
$89$	−3.87589	−0.410843	−0.205422	+	0.978674i	$0.565857\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.23813	0.854254
$94$	0	0
$95$	17.4491	1.79024
$96$	0	0
$97$	−2.10351	−0.213579	−0.106790	−	0.994282i	$-0.534057\pi$
$97$	−2.10351	−0.213579	−0.106790	+	0.994282i	$0.534057\pi$
$98$	0	0
$99$	−1.63513	−0.164336
$100$	0	0
$101$	−12.8092	−1.27456	−0.637281	−	0.770632i	$-0.719941\pi$
$101$	−12.8092	−1.27456	−0.637281	+	0.770632i	$0.719941\pi$
$102$	0	0
$103$	−2.23897	−0.220612	−0.110306	−	0.993898i	$-0.535183\pi$
$103$	−2.23897	−0.220612	−0.110306	+	0.993898i	$0.535183\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.60300	0.735010	0.367505	−	0.930022i	$-0.380212\pi$
$107$	7.60300	0.735010	0.367505	+	0.930022i	$0.380212\pi$
$108$	0	0
$109$	7.56337	0.724440	0.362220	−	0.932093i	$-0.382019\pi$
$109$	7.56337	0.724440	0.362220	+	0.932093i	$0.382019\pi$
$110$	0	0
$111$	−17.4742	−1.65858
$112$	0	0
$113$	−3.17816	−0.298976	−0.149488	−	0.988764i	$-0.547763\pi$
$113$	−3.17816	−0.298976	−0.149488	+	0.988764i	$0.547763\pi$
$114$	0	0
$115$	18.8492	1.75770
$116$	0	0
$117$	6.70172	0.619574
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	24.1837	2.18057
$124$	0	0
$125$	−19.4667	−1.74115
$126$	0	0
$127$	−5.72640	−0.508136	−0.254068	−	0.967186i	$-0.581769\pi$
$127$	−5.72640	−0.508136	−0.254068	+	0.967186i	$0.581769\pi$
$128$	0	0
$129$	−4.00735	−0.352828
$130$	0	0
$131$	0.0240271	0.00209926	0.00104963	−	0.999999i	$-0.499666\pi$
$131$	0.0240271	0.00209926	0.00104963	+	0.999999i	$0.499666\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−11.3892	−0.980230
$136$	0	0
$137$	3.83915	0.328001	0.164000	−	0.986460i	$-0.447560\pi$
$137$	3.83915	0.328001	0.164000	+	0.986460i	$0.447560\pi$
$138$	0	0
$139$	−3.59639	−0.305042	−0.152521	−	0.988300i	$-0.548739\pi$
$139$	−3.59639	−0.305042	−0.152521	+	0.988300i	$0.548739\pi$
$140$	0	0
$141$	16.5770	1.39604
$142$	0	0
$143$	−4.09860	−0.342742
$144$	0	0
$145$	−30.2176	−2.50943
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.5456	−1.02777	−0.513886	−	0.857858i	$-0.671795\pi$
$149$	−12.5456	−1.02777	−0.513886	+	0.857858i	$0.671795\pi$
$150$	0	0
$151$	0.591093	0.0481025	0.0240512	−	0.999711i	$-0.492344\pi$
$151$	0.591093	0.0481025	0.0240512	+	0.999711i	$0.492344\pi$
$152$	0	0
$153$	−1.34857	−0.109026
$154$	0	0
$155$	14.8309	1.19125
$156$	0	0
$157$	−15.2795	−1.21943	−0.609717	−	0.792619i	$-0.708717\pi$
$157$	−15.2795	−1.21943	−0.609717	+	0.792619i	$0.708717\pi$
$158$	0	0
$159$	8.47169	0.671848
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.30282	0.180370	0.0901852	−	0.995925i	$-0.471254\pi$
$163$	2.30282	0.180370	0.0901852	+	0.995925i	$0.471254\pi$
$164$	0	0
$165$	−8.34453	−0.649621
$166$	0	0
$167$	4.60081	0.356021	0.178011	−	0.984029i	$-0.443034\pi$
$167$	4.60081	0.356021	0.178011	+	0.984029i	$0.443034\pi$
$168$	0	0
$169$	3.79851	0.292193
$170$	0	0
$171$	−7.36126	−0.562930
$172$	0	0
$173$	18.0515	1.37243	0.686215	−	0.727399i	$-0.259271\pi$
$173$	18.0515	1.37243	0.686215	+	0.727399i	$0.259271\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−20.3494	−1.52955
$178$	0	0
$179$	−15.5109	−1.15934	−0.579670	−	0.814851i	$-0.696818\pi$
$179$	−15.5109	−1.15934	−0.579670	+	0.814851i	$0.696818\pi$
$180$	0	0
$181$	6.64371	0.493823	0.246912	−	0.969038i	$-0.420584\pi$
$181$	6.64371	0.493823	0.246912	+	0.969038i	$0.420584\pi$
$182$	0	0
$183$	18.0943	1.33757
$184$	0	0
$185$	−31.4585	−2.31287
$186$	0	0
$187$	0.824752	0.0603118
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.94524	−0.357825	−0.178912	−	0.983865i	$-0.557258\pi$
$191$	−4.94524	−0.357825	−0.178912	+	0.983865i	$0.557258\pi$
$192$	0	0
$193$	5.37008	0.386547	0.193273	−	0.981145i	$-0.438090\pi$
$193$	5.37008	0.386547	0.193273	+	0.981145i	$0.438090\pi$
$194$	0	0
$195$	34.2009	2.44918
$196$	0	0
$197$	−22.0487	−1.57091	−0.785454	−	0.618921i	$-0.787570\pi$
$197$	−22.0487	−1.57091	−0.785454	+	0.618921i	$0.787570\pi$
$198$	0	0
$199$	−12.4218	−0.880558	−0.440279	−	0.897861i	$-0.645121\pi$
$199$	−12.4218	−0.880558	−0.440279	+	0.897861i	$0.645121\pi$
$200$	0	0
$201$	20.3649	1.43643
$202$	0	0
$203$	0	0
$204$	0	0
$205$	43.5374	3.04079
$206$	0	0
$207$	−7.95194	−0.552698
$208$	0	0
$209$	4.50196	0.311407
$210$	0	0
$211$	26.7445	1.84117	0.920584	−	0.390545i	$-0.127713\pi$
$211$	26.7445	1.84117	0.920584	+	0.390545i	$0.127713\pi$
$212$	0	0
$213$	28.9766	1.98545
$214$	0	0
$215$	−7.21436	−0.492015
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−13.3342	−0.901045
$220$	0	0
$221$	−3.38033	−0.227385
$222$	0	0
$223$	−1.79215	−0.120011	−0.0600055	−	0.998198i	$-0.519112\pi$
$223$	−1.79215	−0.120011	−0.0600055	+	0.998198i	$0.519112\pi$
$224$	0	0
$225$	16.3881	1.09254
$226$	0	0
$227$	−17.6173	−1.16930	−0.584649	−	0.811286i	$-0.698768\pi$
$227$	−17.6173	−1.16930	−0.584649	+	0.811286i	$0.698768\pi$
$228$	0	0
$229$	5.25609	0.347332	0.173666	−	0.984805i	$-0.444439\pi$
$229$	5.25609	0.347332	0.173666	+	0.984805i	$0.444439\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.7069	−0.897967	−0.448984	−	0.893540i	$-0.648214\pi$
$233$	−13.7069	−0.897967	−0.448984	+	0.893540i	$0.648214\pi$
$234$	0	0
$235$	29.8433	1.94676
$236$	0	0
$237$	1.86962	0.121445
$238$	0	0
$239$	−8.17883	−0.529045	−0.264522	−	0.964380i	$-0.585214\pi$
$239$	−8.17883	−0.529045	−0.264522	+	0.964380i	$0.585214\pi$
$240$	0	0
$241$	−11.3061	−0.728290	−0.364145	−	0.931342i	$-0.618639\pi$
$241$	−11.3061	−0.728290	−0.364145	+	0.931342i	$0.618639\pi$
$242$	0	0
$243$	15.3657	0.985713
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−18.4517	−1.17405
$248$	0	0
$249$	17.6480	1.11840
$250$	0	0
$251$	19.9903	1.26178	0.630889	−	0.775873i	$-0.282691\pi$
$251$	19.9903	1.26178	0.630889	+	0.775873i	$0.282691\pi$
$252$	0	0
$253$	4.86320	0.305747
$254$	0	0
$255$	−6.88217	−0.430978
$256$	0	0
$257$	−6.13629	−0.382771	−0.191386	−	0.981515i	$-0.561298\pi$
$257$	−6.13629	−0.382771	−0.191386	+	0.981515i	$0.561298\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	12.7479	0.789076
$262$	0	0
$263$	22.3223	1.37645	0.688227	−	0.725495i	$-0.258389\pi$
$263$	22.3223	1.37645	0.688227	+	0.725495i	$0.258389\pi$
$264$	0	0
$265$	15.2514	0.936888
$266$	0	0
$267$	8.34453	0.510677
$268$	0	0
$269$	3.43592	0.209492	0.104746	−	0.994499i	$-0.466597\pi$
$269$	3.43592	0.209492	0.104746	+	0.994499i	$0.466597\pi$
$270$	0	0
$271$	−17.8410	−1.08376	−0.541880	−	0.840456i	$-0.682287\pi$
$271$	−17.8410	−1.08376	−0.541880	+	0.840456i	$0.682287\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10.0225	−0.604380
$276$	0	0
$277$	16.9426	1.01798	0.508990	−	0.860772i	$-0.330019\pi$
$277$	16.9426	1.01798	0.508990	+	0.860772i	$0.330019\pi$
$278$	0	0
$279$	−6.25675	−0.374582
$280$	0	0
$281$	24.2080	1.44413	0.722066	−	0.691825i	$-0.243193\pi$
$281$	24.2080	1.44413	0.722066	+	0.691825i	$0.243193\pi$
$282$	0	0
$283$	2.24541	0.133475	0.0667377	−	0.997771i	$-0.478741\pi$
$283$	2.24541	0.133475	0.0667377	+	0.997771i	$0.478741\pi$
$284$	0	0
$285$	−37.5667	−2.22526
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.3198	−0.959987
$290$	0	0
$291$	4.52872	0.265478
$292$	0	0
$293$	−0.635515	−0.0371272	−0.0185636	−	0.999828i	$-0.505909\pi$
$293$	−0.635515	−0.0371272	−0.0185636	+	0.999828i	$0.505909\pi$
$294$	0	0
$295$	−36.6346	−2.13295
$296$	0	0
$297$	−2.93849	−0.170508
$298$	0	0
$299$	−19.9323	−1.15271
$300$	0	0
$301$	0	0
$302$	0	0
$303$	27.5773	1.58428
$304$	0	0
$305$	32.5748	1.86523
$306$	0	0
$307$	17.7490	1.01299	0.506493	−	0.862244i	$-0.330941\pi$
$307$	17.7490	1.01299	0.506493	+	0.862244i	$0.330941\pi$
$308$	0	0
$309$	4.82034	0.274220
$310$	0	0
$311$	−11.2429	−0.637525	−0.318763	−	0.947835i	$-0.603267\pi$
$311$	−11.2429	−0.637525	−0.318763	+	0.947835i	$0.603267\pi$
$312$	0	0
$313$	−4.25546	−0.240533	−0.120266	−	0.992742i	$-0.538375\pi$
$313$	−4.25546	−0.240533	−0.120266	+	0.992742i	$0.538375\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.2336	1.19260	0.596299	−	0.802763i	$-0.296637\pi$
$317$	21.2336	1.19260	0.596299	+	0.802763i	$0.296637\pi$
$318$	0	0
$319$	−7.79629	−0.436509
$320$	0	0
$321$	−16.3688	−0.913615
$322$	0	0
$323$	3.71300	0.206597
$324$	0	0
$325$	41.0783	2.27861
$326$	0	0
$327$	−16.2834	−0.900477
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−23.4568	−1.28930	−0.644652	−	0.764476i	$-0.722998\pi$
$331$	−23.4568	−1.28930	−0.644652	+	0.764476i	$0.722998\pi$
$332$	0	0
$333$	13.2714	0.727270
$334$	0	0
$335$	36.6626	2.00309
$336$	0	0
$337$	−34.7640	−1.89372	−0.946859	−	0.321650i	$-0.895762\pi$
$337$	−34.7640	−1.89372	−0.946859	+	0.321650i	$0.895762\pi$
$338$	0	0
$339$	6.84236	0.371626
$340$	0	0
$341$	3.82646	0.207215
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−40.5811	−2.18481
$346$	0	0
$347$	0.689571	0.0370181	0.0185090	−	0.999829i	$-0.494108\pi$
$347$	0.689571	0.0370181	0.0185090	+	0.999829i	$0.494108\pi$
$348$	0	0
$349$	19.9995	1.07055	0.535275	−	0.844678i	$-0.320208\pi$
$349$	19.9995	1.07055	0.535275	+	0.844678i	$0.320208\pi$
$350$	0	0
$351$	12.0437	0.642844
$352$	0	0
$353$	3.57359	0.190203	0.0951014	−	0.995468i	$-0.469682\pi$
$353$	3.57359	0.190203	0.0951014	+	0.995468i	$0.469682\pi$
$354$	0	0
$355$	52.1661	2.76869
$356$	0	0
$357$	0	0
$358$	0	0
$359$	31.6809	1.67205	0.836026	−	0.548690i	$-0.184873\pi$
$359$	31.6809	1.67205	0.836026	+	0.548690i	$0.184873\pi$
$360$	0	0
$361$	1.26762	0.0667169
$362$	0	0
$363$	−2.15293	−0.113000
$364$	0	0
$365$	−24.0054	−1.25650
$366$	0	0
$367$	17.6491	0.921277	0.460638	−	0.887588i	$-0.347620\pi$
$367$	17.6491	0.921277	0.460638	+	0.887588i	$0.347620\pi$
$368$	0	0
$369$	−18.3672	−0.956158
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−24.0524	−1.24539	−0.622694	−	0.782465i	$-0.713962\pi$
$373$	−24.0524	−1.24539	−0.622694	+	0.782465i	$0.713962\pi$
$374$	0	0
$375$	41.9105	2.16425
$376$	0	0
$377$	31.9539	1.64571
$378$	0	0
$379$	−5.73645	−0.294662	−0.147331	−	0.989087i	$-0.547068\pi$
$379$	−5.73645	−0.294662	−0.147331	+	0.989087i	$0.547068\pi$
$380$	0	0
$381$	12.3286	0.631611
$382$	0	0
$383$	9.45929	0.483347	0.241674	−	0.970358i	$-0.422304\pi$
$383$	9.45929	0.483347	0.241674	+	0.970358i	$0.422304\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.04353	0.154711
$388$	0	0
$389$	16.3482	0.828887	0.414443	−	0.910075i	$-0.363976\pi$
$389$	16.3482	0.828887	0.414443	+	0.910075i	$0.363976\pi$
$390$	0	0
$391$	4.01093	0.202841
$392$	0	0
$393$	−0.0517288	−0.00260937
$394$	0	0
$395$	3.36584	0.169354
$396$	0	0
$397$	31.2208	1.56693	0.783464	−	0.621437i	$-0.213451\pi$
$397$	31.2208	1.56693	0.783464	+	0.621437i	$0.213451\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.6480	0.681550	0.340775	−	0.940145i	$-0.389311\pi$
$401$	13.6480	0.681550	0.340775	+	0.940145i	$0.389311\pi$
$402$	0	0
$403$	−15.6831	−0.781233
$404$	0	0
$405$	43.5330	2.16317
$406$	0	0
$407$	−8.11646	−0.402318
$408$	0	0
$409$	13.6468	0.674790	0.337395	−	0.941363i	$-0.390454\pi$
$409$	13.6468	0.674790	0.337395	+	0.941363i	$0.390454\pi$
$410$	0	0
$411$	−8.26543	−0.407704
$412$	0	0
$413$	0	0
$414$	0	0
$415$	31.7714	1.55960
$416$	0	0
$417$	7.74279	0.379166
$418$	0	0
$419$	−13.1381	−0.641840	−0.320920	−	0.947106i	$-0.603992\pi$
$419$	−13.1381	−0.641840	−0.320920	+	0.947106i	$0.603992\pi$
$420$	0	0
$421$	10.6056	0.516883	0.258441	−	0.966027i	$-0.416791\pi$
$421$	10.6056	0.516883	0.258441	+	0.966027i	$0.416791\pi$
$422$	0	0
$423$	−12.5900	−0.612148
$424$	0	0
$425$	−8.26608	−0.400964
$426$	0	0
$427$	0	0
$428$	0	0
$429$	8.82401	0.426027
$430$	0	0
$431$	29.5575	1.42373	0.711867	−	0.702315i	$-0.247850\pi$
$431$	29.5575	1.42373	0.711867	+	0.702315i	$0.247850\pi$
$432$	0	0
$433$	32.7665	1.57466	0.787329	−	0.616534i	$-0.211464\pi$
$433$	32.7665	1.57466	0.787329	+	0.616534i	$0.211464\pi$
$434$	0	0
$435$	65.0564	3.11922
$436$	0	0
$437$	21.8939	1.04733
$438$	0	0
$439$	38.5068	1.83783	0.918914	−	0.394459i	$-0.129068\pi$
$439$	38.5068	1.83783	0.918914	+	0.394459i	$0.129068\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−31.2823	−1.48627	−0.743134	−	0.669142i	$-0.766662\pi$
$443$	−31.2823	−1.48627	−0.743134	+	0.669142i	$0.766662\pi$
$444$	0	0
$445$	15.0225	0.712135
$446$	0	0
$447$	27.0098	1.27752
$448$	0	0
$449$	−8.82808	−0.416623	−0.208311	−	0.978063i	$-0.566797\pi$
$449$	−8.82808	−0.416623	−0.208311	+	0.978063i	$0.566797\pi$
$450$	0	0
$451$	11.2329	0.528936
$452$	0	0
$453$	−1.27259	−0.0597913
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.1378	−0.895228	−0.447614	−	0.894227i	$-0.647726\pi$
$457$	−19.1378	−0.895228	−0.447614	+	0.894227i	$0.647726\pi$
$458$	0	0
$459$	−2.42352	−0.113120
$460$	0	0
$461$	20.9755	0.976926	0.488463	−	0.872584i	$-0.337558\pi$
$461$	20.9755	0.976926	0.488463	+	0.872584i	$0.337558\pi$
$462$	0	0
$463$	32.4558	1.50835	0.754174	−	0.656674i	$-0.228037\pi$
$463$	32.4558	1.50835	0.754174	+	0.656674i	$0.228037\pi$
$464$	0	0
$465$	−31.9301	−1.48072
$466$	0	0
$467$	4.99179	0.230993	0.115496	−	0.993308i	$-0.463154\pi$
$467$	4.99179	0.230993	0.115496	+	0.993308i	$0.463154\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	32.8957	1.51575
$472$	0	0
$473$	−1.86134	−0.0855847
$474$	0	0
$475$	−45.1209	−2.07029
$476$	0	0
$477$	−6.43414	−0.294599
$478$	0	0
$479$	20.8655	0.953368	0.476684	−	0.879075i	$-0.341839\pi$
$479$	20.8655	0.953368	0.476684	+	0.879075i	$0.341839\pi$
$480$	0	0
$481$	33.2661	1.51680
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.15297	0.370207
$486$	0	0
$487$	23.9012	1.08307	0.541533	−	0.840680i	$-0.317844\pi$
$487$	23.9012	1.08307	0.541533	+	0.840680i	$0.317844\pi$
$488$	0	0
$489$	−4.95781	−0.224200
$490$	0	0
$491$	−19.3897	−0.875044	−0.437522	−	0.899208i	$-0.644144\pi$
$491$	−19.3897	−0.875044	−0.437522	+	0.899208i	$0.644144\pi$
$492$	0	0
$493$	−6.43001	−0.289593
$494$	0	0
$495$	6.33756	0.284852
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0.100167	0.00448408	0.00224204	−	0.999997i	$-0.499286\pi$
$499$	0.100167	0.00448408	0.00224204	+	0.999997i	$0.499286\pi$
$500$	0	0
$501$	−9.90523	−0.442533
$502$	0	0
$503$	−14.2263	−0.634317	−0.317159	−	0.948372i	$-0.602729\pi$
$503$	−14.2263	−0.634317	−0.317159	+	0.948372i	$0.602729\pi$
$504$	0	0
$505$	49.6470	2.20926
$506$	0	0
$507$	−8.17794	−0.363195
$508$	0	0
$509$	8.67585	0.384550	0.192275	−	0.981341i	$-0.438413\pi$
$509$	8.67585	0.384550	0.192275	+	0.981341i	$0.438413\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−13.2289	−0.584072
$514$	0	0
$515$	8.67798	0.382397
$516$	0	0
$517$	7.69973	0.338634
$518$	0	0
$519$	−38.8637	−1.70593
$520$	0	0
$521$	−11.0313	−0.483288	−0.241644	−	0.970365i	$-0.577687\pi$
$521$	−11.0313	−0.483288	−0.241644	+	0.970365i	$0.577687\pi$
$522$	0	0
$523$	11.4402	0.500243	0.250122	−	0.968214i	$-0.419529\pi$
$523$	11.4402	0.500243	0.250122	+	0.968214i	$0.419529\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.15588	0.137472
$528$	0	0
$529$	0.650700	0.0282913
$530$	0	0
$531$	15.4551	0.670694
$532$	0	0
$533$	−46.0391	−1.99418
$534$	0	0
$535$	−29.4684	−1.27403
$536$	0	0
$537$	33.3940	1.44106
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1.20512	0.0518122	0.0259061	−	0.999664i	$-0.491753\pi$
$541$	1.20512	0.0518122	0.0259061	+	0.999664i	$0.491753\pi$
$542$	0	0
$543$	−14.3035	−0.613821
$544$	0	0
$545$	−29.3148	−1.25571
$546$	0	0
$547$	−5.64802	−0.241492	−0.120746	−	0.992683i	$-0.538529\pi$
$547$	−5.64802	−0.241492	−0.120746	+	0.992683i	$0.538529\pi$
$548$	0	0
$549$	−13.7424	−0.586509
$550$	0	0
$551$	−35.0986	−1.49525
$552$	0	0
$553$	0	0
$554$	0	0
$555$	67.7281	2.87490
$556$	0	0
$557$	−6.89754	−0.292258	−0.146129	−	0.989266i	$-0.546681\pi$
$557$	−6.89754	−0.292258	−0.146129	+	0.989266i	$0.546681\pi$
$558$	0	0
$559$	7.62890	0.322668
$560$	0	0
$561$	−1.77564	−0.0749674
$562$	0	0
$563$	−1.52144	−0.0641211	−0.0320605	−	0.999486i	$-0.510207\pi$
$563$	−1.52144	−0.0641211	−0.0320605	+	0.999486i	$0.510207\pi$
$564$	0	0
$565$	12.3182	0.518230
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.5468	1.19674	0.598372	−	0.801219i	$-0.295815\pi$
$569$	28.5468	1.19674	0.598372	+	0.801219i	$0.295815\pi$
$570$	0	0
$571$	−37.5519	−1.57150	−0.785749	−	0.618545i	$-0.787722\pi$
$571$	−37.5519	−1.57150	−0.785749	+	0.618545i	$0.787722\pi$
$572$	0	0
$573$	10.6468	0.444775
$574$	0	0
$575$	−48.7415	−2.03266
$576$	0	0
$577$	12.5401	0.522051	0.261025	−	0.965332i	$-0.415939\pi$
$577$	12.5401	0.522051	0.261025	+	0.965332i	$0.415939\pi$
$578$	0	0
$579$	−11.5614	−0.480477
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3.93495	0.162969
$584$	0	0
$585$	−25.9751	−1.07394
$586$	0	0
$587$	33.6526	1.38899	0.694496	−	0.719496i	$-0.255627\pi$
$587$	33.6526	1.38899	0.694496	+	0.719496i	$0.255627\pi$
$588$	0	0
$589$	17.2266	0.709809
$590$	0	0
$591$	47.4695	1.95263
$592$	0	0
$593$	−24.4214	−1.00287	−0.501434	−	0.865196i	$-0.667194\pi$
$593$	−24.4214	−1.00287	−0.501434	+	0.865196i	$0.667194\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	26.7433	1.09453
$598$	0	0
$599$	−5.56936	−0.227558	−0.113779	−	0.993506i	$-0.536296\pi$
$599$	−5.56936	−0.227558	−0.113779	+	0.993506i	$0.536296\pi$
$600$	0	0
$601$	−10.8079	−0.440864	−0.220432	−	0.975402i	$-0.570747\pi$
$601$	−10.8079	−0.440864	−0.220432	+	0.975402i	$0.570747\pi$
$602$	0	0
$603$	−15.4669	−0.629860
$604$	0	0
$605$	−3.87589	−0.157577
$606$	0	0
$607$	10.5106	0.426611	0.213305	−	0.976986i	$-0.431577\pi$
$607$	10.5106	0.426611	0.213305	+	0.976986i	$0.431577\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−31.5581	−1.27670
$612$	0	0
$613$	−23.4044	−0.945293	−0.472647	−	0.881252i	$-0.656701\pi$
$613$	−23.4044	−0.945293	−0.472647	+	0.881252i	$0.656701\pi$
$614$	0	0
$615$	−93.7332	−3.77969
$616$	0	0
$617$	−27.4432	−1.10482	−0.552410	−	0.833572i	$-0.686292\pi$
$617$	−27.4432	−1.10482	−0.552410	+	0.833572i	$0.686292\pi$
$618$	0	0
$619$	−8.37072	−0.336448	−0.168224	−	0.985749i	$-0.553803\pi$
$619$	−8.37072	−0.336448	−0.168224	+	0.985749i	$0.553803\pi$
$620$	0	0
$621$	−14.2904	−0.573455
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.3382	1.01353
$626$	0	0
$627$	−9.69242	−0.387078
$628$	0	0
$629$	−6.69406	−0.266910
$630$	0	0
$631$	40.8816	1.62747	0.813736	−	0.581235i	$-0.197430\pi$
$631$	40.8816	1.62747	0.813736	+	0.581235i	$0.197430\pi$
$632$	0	0
$633$	−57.5792	−2.28857
$634$	0	0
$635$	22.1949	0.880777
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−22.0074	−0.870598
$640$	0	0
$641$	−28.8987	−1.14143	−0.570714	−	0.821149i	$-0.693334\pi$
$641$	−28.8987	−1.14143	−0.570714	+	0.821149i	$0.693334\pi$
$642$	0	0
$643$	44.9484	1.77259	0.886296	−	0.463120i	$-0.153270\pi$
$643$	44.9484	1.77259	0.886296	+	0.463120i	$0.153270\pi$
$644$	0	0
$645$	15.5320	0.611574
$646$	0	0
$647$	−1.45440	−0.0571784	−0.0285892	−	0.999591i	$-0.509101\pi$
$647$	−1.45440	−0.0571784	−0.0285892	+	0.999591i	$0.509101\pi$
$648$	0	0
$649$	−9.45193	−0.371021
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.89313	0.191483	0.0957415	−	0.995406i	$-0.469478\pi$
$653$	4.89313	0.191483	0.0957415	+	0.995406i	$0.469478\pi$
$654$	0	0
$655$	−0.0931263	−0.00363875
$656$	0	0
$657$	10.1272	0.395099
$658$	0	0
$659$	−14.2035	−0.553291	−0.276646	−	0.960972i	$-0.589223\pi$
$659$	−14.2035	−0.553291	−0.276646	+	0.960972i	$0.589223\pi$
$660$	0	0
$661$	−33.4970	−1.30288	−0.651442	−	0.758698i	$-0.725836\pi$
$661$	−33.4970	−1.30288	−0.651442	+	0.758698i	$0.725836\pi$
$662$	0	0
$663$	7.27762	0.282639
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−37.9149	−1.46807
$668$	0	0
$669$	3.85837	0.149173
$670$	0	0
$671$	8.40447	0.324451
$672$	0	0
$673$	−50.4763	−1.94572	−0.972859	−	0.231398i	$-0.925670\pi$
$673$	−50.4763	−1.94572	−0.972859	+	0.231398i	$0.925670\pi$
$674$	0	0
$675$	29.4510	1.13357
$676$	0	0
$677$	−41.4961	−1.59482	−0.797412	−	0.603436i	$-0.793798\pi$
$677$	−41.4961	−1.59482	−0.797412	+	0.603436i	$0.793798\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	37.9288	1.45343
$682$	0	0
$683$	0.914566	0.0349949	0.0174975	−	0.999847i	$-0.494430\pi$
$683$	0.914566	0.0349949	0.0174975	+	0.999847i	$0.494430\pi$
$684$	0	0
$685$	−14.8801	−0.568540
$686$	0	0
$687$	−11.3160	−0.431733
$688$	0	0
$689$	−16.1278	−0.614419
$690$	0	0
$691$	33.6540	1.28026	0.640130	−	0.768267i	$-0.278881\pi$
$691$	33.6540	1.28026	0.640130	+	0.768267i	$0.278881\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.9392	0.528744
$696$	0	0
$697$	9.26435	0.350912
$698$	0	0
$699$	29.5100	1.11617
$700$	0	0
$701$	−19.9539	−0.753649	−0.376825	−	0.926285i	$-0.622984\pi$
$701$	−19.9539	−0.753649	−0.376825	+	0.926285i	$0.622984\pi$
$702$	0	0
$703$	−36.5400	−1.37813
$704$	0	0
$705$	−64.2507	−2.41982
$706$	0	0
$707$	0	0
$708$	0	0
$709$	17.9062	0.672481	0.336240	−	0.941776i	$-0.390845\pi$
$709$	17.9062	0.672481	0.336240	+	0.941776i	$0.390845\pi$
$710$	0	0
$711$	−1.41995	−0.0532523
$712$	0	0
$713$	18.6089	0.696907
$714$	0	0
$715$	15.8857	0.594092
$716$	0	0
$717$	17.6085	0.657601
$718$	0	0
$719$	2.99817	0.111813	0.0559064	−	0.998436i	$-0.482195\pi$
$719$	2.99817	0.111813	0.0559064	+	0.998436i	$0.482195\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	24.3413	0.905263
$724$	0	0
$725$	78.1384	2.90199
$726$	0	0
$727$	29.4088	1.09071	0.545355	−	0.838205i	$-0.316395\pi$
$727$	29.4088	1.09071	0.545355	+	0.838205i	$0.316395\pi$
$728$	0	0
$729$	0.613791	0.0227330
$730$	0	0
$731$	−1.53515	−0.0567794
$732$	0	0
$733$	−1.39138	−0.0513918	−0.0256959	−	0.999670i	$-0.508180\pi$
$733$	−1.39138	−0.0513918	−0.0256959	+	0.999670i	$0.508180\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.45914	0.348432
$738$	0	0
$739$	−9.54756	−0.351213	−0.175606	−	0.984460i	$-0.556189\pi$
$739$	−9.54756	−0.351213	−0.175606	+	0.984460i	$0.556189\pi$
$740$	0	0
$741$	39.7253	1.45935
$742$	0	0
$743$	15.7641	0.578328	0.289164	−	0.957280i	$-0.406623\pi$
$743$	15.7641	0.578328	0.289164	+	0.957280i	$0.406623\pi$
$744$	0	0
$745$	48.6252	1.78149
$746$	0	0
$747$	−13.4034	−0.490406
$748$	0	0
$749$	0	0
$750$	0	0
$751$	43.1559	1.57478	0.787391	−	0.616455i	$-0.211432\pi$
$751$	43.1559	1.57478	0.787391	+	0.616455i	$0.211432\pi$
$752$	0	0
$753$	−43.0378	−1.56839
$754$	0	0
$755$	−2.29101	−0.0833785
$756$	0	0
$757$	−36.7539	−1.33584	−0.667922	−	0.744231i	$-0.732816\pi$
$757$	−36.7539	−1.33584	−0.667922	+	0.744231i	$0.732816\pi$
$758$	0	0
$759$	−10.4701	−0.380042
$760$	0	0
$761$	34.6094	1.25459	0.627296	−	0.778781i	$-0.284162\pi$
$761$	34.6094	1.25459	0.627296	+	0.778781i	$0.284162\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	5.22692	0.188980
$766$	0	0
$767$	38.7397	1.39881
$768$	0	0
$769$	−28.4872	−1.02728	−0.513638	−	0.858007i	$-0.671703\pi$
$769$	−28.4872	−1.02728	−0.513638	+	0.858007i	$0.671703\pi$
$770$	0	0
$771$	13.2110	0.475783
$772$	0	0
$773$	11.5635	0.415910	0.207955	−	0.978138i	$-0.433319\pi$
$773$	11.5635	0.415910	0.207955	+	0.978138i	$0.433319\pi$
$774$	0	0
$775$	−38.3508	−1.37760
$776$	0	0
$777$	0	0
$778$	0	0
$779$	50.5700	1.81186
$780$	0	0
$781$	13.4591	0.481606
$782$	0	0
$783$	22.9093	0.818711
$784$	0	0
$785$	59.2215	2.11371
$786$	0	0
$787$	4.02977	0.143646	0.0718229	−	0.997417i	$-0.477118\pi$
$787$	4.02977	0.143646	0.0718229	+	0.997417i	$0.477118\pi$
$788$	0	0
$789$	−48.0585	−1.71093
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−34.4465	−1.22323
$794$	0	0
$795$	−32.8353	−1.16455
$796$	0	0
$797$	5.65757	0.200401	0.100201	−	0.994967i	$-0.468052\pi$
$797$	5.65757	0.200401	0.100201	+	0.994967i	$0.468052\pi$
$798$	0	0
$799$	6.35036	0.224660
$800$	0	0
$801$	−6.33756	−0.223927
$802$	0	0
$803$	−6.19352	−0.218565
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.39732	−0.260398
$808$	0	0
$809$	17.2095	0.605052	0.302526	−	0.953141i	$-0.402170\pi$
$809$	17.2095	0.605052	0.302526	+	0.953141i	$0.402170\pi$
$810$	0	0
$811$	33.3821	1.17221	0.586103	−	0.810237i	$-0.300662\pi$
$811$	33.3821	1.17221	0.586103	+	0.810237i	$0.300662\pi$
$812$	0	0
$813$	38.4104	1.34711
$814$	0	0
$815$	−8.92546	−0.312645
$816$	0	0
$817$	−8.37969	−0.293168
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0.263189	0.00918537	0.00459269	−	0.999989i	$-0.498538\pi$
$821$	0.263189	0.00918537	0.00459269	+	0.999989i	$0.498538\pi$
$822$	0	0
$823$	−14.8315	−0.516995	−0.258497	−	0.966012i	$-0.583227\pi$
$823$	−14.8315	−0.516995	−0.258497	+	0.966012i	$0.583227\pi$
$824$	0	0
$825$	21.5778	0.751243
$826$	0	0
$827$	−38.1437	−1.32639	−0.663194	−	0.748448i	$-0.730799\pi$
$827$	−38.1437	−1.32639	−0.663194	+	0.748448i	$0.730799\pi$
$828$	0	0
$829$	−10.4064	−0.361429	−0.180715	−	0.983536i	$-0.557841\pi$
$829$	−10.4064	−0.361429	−0.180715	+	0.983536i	$0.557841\pi$
$830$	0	0
$831$	−36.4762	−1.26535
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−17.8322	−0.617109
$836$	0	0
$837$	−11.2440	−0.388650
$838$	0	0
$839$	−35.4338	−1.22331	−0.611655	−	0.791125i	$-0.709496\pi$
$839$	−35.4338	−1.22331	−0.611655	+	0.791125i	$0.709496\pi$
$840$	0	0
$841$	31.7822	1.09594
$842$	0	0
$843$	−52.1183	−1.79505
$844$	0	0
$845$	−14.7226	−0.506473
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−4.83421	−0.165910
$850$	0	0
$851$	−39.4720	−1.35308
$852$	0	0
$853$	10.5950	0.362765	0.181382	−	0.983413i	$-0.441943\pi$
$853$	10.5950	0.362765	0.181382	+	0.983413i	$0.441943\pi$
$854$	0	0
$855$	28.5314	0.975755
$856$	0	0
$857$	−42.4041	−1.44850	−0.724248	−	0.689540i	$-0.757813\pi$
$857$	−42.4041	−1.44850	−0.724248	+	0.689540i	$0.757813\pi$
$858$	0	0
$859$	−7.47204	−0.254943	−0.127471	−	0.991842i	$-0.540686\pi$
$859$	−7.47204	−0.254943	−0.127471	+	0.991842i	$0.540686\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30.5892	−1.04127	−0.520635	−	0.853779i	$-0.674305\pi$
$863$	−30.5892	−1.04127	−0.520635	+	0.853779i	$0.674305\pi$
$864$	0	0
$865$	−69.9656	−2.37890
$866$	0	0
$867$	35.1354	1.19326
$868$	0	0
$869$	0.868405	0.0294586
$870$	0	0
$871$	−38.7692	−1.31364
$872$	0	0
$873$	−3.43950	−0.116410
$874$	0	0
$875$	0	0
$876$	0	0
$877$	51.7674	1.74806	0.874030	−	0.485872i	$-0.161498\pi$
$877$	51.7674	1.74806	0.874030	+	0.485872i	$0.161498\pi$
$878$	0	0
$879$	1.36822	0.0461490
$880$	0	0
$881$	41.8750	1.41081	0.705403	−	0.708807i	$-0.250766\pi$
$881$	41.8750	1.41081	0.705403	+	0.708807i	$0.250766\pi$
$882$	0	0
$883$	38.7098	1.30269	0.651344	−	0.758782i	$-0.274205\pi$
$883$	38.7098	1.30269	0.651344	+	0.758782i	$0.274205\pi$
$884$	0	0
$885$	78.8719	2.65125
$886$	0	0
$887$	44.1008	1.48076	0.740380	−	0.672188i	$-0.234645\pi$
$887$	44.1008	1.48076	0.740380	+	0.672188i	$0.234645\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	11.2317	0.376277
$892$	0	0
$893$	34.6639	1.15998
$894$	0	0
$895$	60.1185	2.00954
$896$	0	0
$897$	42.9129	1.43282
$898$	0	0
$899$	−29.8322	−0.994961
$900$	0	0
$901$	3.24536	0.108118
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−25.7503	−0.855968
$906$	0	0
$907$	37.5463	1.24670	0.623351	−	0.781942i	$-0.285771\pi$
$907$	37.5463	1.24670	0.623351	+	0.781942i	$0.285771\pi$
$908$	0	0
$909$	−20.9446	−0.694689
$910$	0	0
$911$	−11.2965	−0.374269	−0.187134	−	0.982334i	$-0.559920\pi$
$911$	−11.2965	−0.374269	−0.187134	+	0.982334i	$0.559920\pi$
$912$	0	0
$913$	8.19720	0.271288
$914$	0	0
$915$	−70.1313	−2.31847
$916$	0	0
$917$	0	0
$918$	0	0
$919$	7.68871	0.253627	0.126814	−	0.991927i	$-0.459525\pi$
$919$	7.68871	0.253627	0.126814	+	0.991927i	$0.459525\pi$
$920$	0	0
$921$	−38.2123	−1.25914
$922$	0	0
$923$	−55.1636	−1.81573
$924$	0	0
$925$	81.3473	2.67468
$926$	0	0
$927$	−3.66099	−0.120243
$928$	0	0
$929$	39.5573	1.29783	0.648916	−	0.760860i	$-0.275223\pi$
$929$	39.5573	1.29783	0.648916	+	0.760860i	$0.275223\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	24.2052	0.792442
$934$	0	0
$935$	−3.19665	−0.104541
$936$	0	0
$937$	−52.4348	−1.71297	−0.856485	−	0.516173i	$-0.827356\pi$
$937$	−52.4348	−1.71297	−0.856485	+	0.516173i	$0.827356\pi$
$938$	0	0
$939$	9.16172	0.298981
$940$	0	0
$941$	49.1101	1.60094	0.800472	−	0.599370i	$-0.204582\pi$
$941$	49.1101	1.60094	0.800472	+	0.599370i	$0.204582\pi$
$942$	0	0
$943$	54.6278	1.77893
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.82953	0.124443	0.0622216	−	0.998062i	$-0.480181\pi$
$947$	3.82953	0.124443	0.0622216	+	0.998062i	$0.480181\pi$
$948$	0	0
$949$	25.3848	0.824024
$950$	0	0
$951$	−45.7145	−1.48240
$952$	0	0
$953$	−45.7248	−1.48117	−0.740585	−	0.671963i	$-0.765452\pi$
$953$	−45.7248	−1.48117	−0.740585	+	0.671963i	$0.765452\pi$
$954$	0	0
$955$	19.1672	0.620236
$956$	0	0
$957$	16.7849	0.542579
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−16.3582	−0.527683
$962$	0	0
$963$	12.4319	0.400611
$964$	0	0
$965$	−20.8138	−0.670021
$966$	0	0
$967$	−40.7847	−1.31155	−0.655773	−	0.754958i	$-0.727657\pi$
$967$	−40.7847	−1.31155	−0.655773	+	0.754958i	$0.727657\pi$
$968$	0	0
$969$	−7.99384	−0.256799
$970$	0	0
$971$	−47.1520	−1.51318	−0.756589	−	0.653890i	$-0.773136\pi$
$971$	−47.1520	−1.51318	−0.756589	+	0.653890i	$0.773136\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−88.4388	−2.83231
$976$	0	0
$977$	−25.3516	−0.811068	−0.405534	−	0.914080i	$-0.632914\pi$
$977$	−25.3516	−0.811068	−0.405534	+	0.914080i	$0.632914\pi$
$978$	0	0
$979$	3.87589	0.123874
$980$	0	0
$981$	12.3671	0.394850
$982$	0	0
$983$	−61.6984	−1.96787	−0.983936	−	0.178520i	$-0.942869\pi$
$983$	−61.6984	−1.96787	−0.983936	+	0.178520i	$0.942869\pi$
$984$	0	0
$985$	85.4584	2.72293
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.05209	−0.287840
$990$	0	0
$991$	−43.8014	−1.39140	−0.695698	−	0.718334i	$-0.744905\pi$
$991$	−43.8014	−1.39140	−0.695698	+	0.718334i	$0.744905\pi$
$992$	0	0
$993$	50.5010	1.60260
$994$	0	0
$995$	48.1455	1.52632
$996$	0	0
$997$	−29.6657	−0.939522	−0.469761	−	0.882794i	$-0.655660\pi$
$997$	−29.6657	−0.939522	−0.469761	+	0.882794i	$0.655660\pi$
$998$	0	0
$999$	23.8501	0.754584

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.df.1.3		10
4.3	odd	2		539.2.a.l.1.4	yes	10
7.6	odd	2	inner	8624.2.a.df.1.8		10
12.11	even	2		4851.2.a.cg.1.8		10
28.3	even	6		539.2.e.o.177.8		20
28.11	odd	6		539.2.e.o.177.7		20
28.19	even	6		539.2.e.o.67.8		20
28.23	odd	6		539.2.e.o.67.7		20
28.27	even	2		539.2.a.l.1.3	✓	10
44.43	even	2		5929.2.a.bv.1.8		10
84.83	odd	2		4851.2.a.cg.1.7		10
308.307	odd	2		5929.2.a.bv.1.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.a.l.1.3	✓	10	28.27	even	2
539.2.a.l.1.4	yes	10	4.3	odd	2
539.2.e.o.67.7		20	28.23	odd	6
539.2.e.o.67.8		20	28.19	even	6
539.2.e.o.177.7		20	28.11	odd	6
539.2.e.o.177.8		20	28.3	even	6
4851.2.a.cg.1.7		10	84.83	odd	2
4851.2.a.cg.1.8		10	12.11	even	2
5929.2.a.bv.1.7		10	308.307	odd	2
5929.2.a.bv.1.8		10	44.43	even	2
8624.2.a.df.1.3		10	1.1	even	1	trivial
8624.2.a.df.1.8		10	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q-2.15293 q^{3} -3.87589 q^{5} +1.63513 q^{9} -1.00000 q^{11} +4.09860 q^{13} +8.34453 q^{15} -0.824752 q^{17} -4.50196 q^{19} -4.86320 q^{23} +10.0225 q^{25} +2.93849 q^{27} +7.79629 q^{29} -3.82646 q^{31} +2.15293 q^{33} +8.11646 q^{37} -8.82401 q^{39} -11.2329 q^{41} +1.86134 q^{43} -6.33756 q^{45} -7.69973 q^{47} +1.77564 q^{51} -3.93495 q^{53} +3.87589 q^{55} +9.69242 q^{57} +9.45193 q^{59} -8.40447 q^{61} -15.8857 q^{65} -9.45914 q^{67} +10.4701 q^{69} -13.4591 q^{71} +6.19352 q^{73} -21.5778 q^{75} -0.868405 q^{79} -11.2317 q^{81} -8.19720 q^{83} +3.19665 q^{85} -16.7849 q^{87} -3.87589 q^{89} +8.23813 q^{93} +17.4491 q^{95} -2.10351 q^{97} -1.63513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 22 q^{9} - 10 q^{11} - 8 q^{15} - 4 q^{23} + 18 q^{25} + 12 q^{29} + 40 q^{37} + 16 q^{39} + 8 q^{43} + 16 q^{53} - 8 q^{57} - 32 q^{65} + 4 q^{67} - 36 q^{71} - 8 q^{79} - 6 q^{81} + 88 q^{85} + 44 q^{93}+ \cdots - 22 q^{99}+O(q^{100}) \) 10 * q + 22 * q^9 - 10 * q^11 - 8 * q^15 - 4 * q^23 + 18 * q^25 + 12 * q^29 + 40 * q^37 + 16 * q^39 + 8 * q^43 + 16 * q^53 - 8 * q^57 - 32 * q^65 + 4 * q^67 - 36 * q^71 - 8 * q^79 - 6 * q^81 + 88 * q^85 + 44 * q^93 + 64 * q^95 - 22 * q^99

Introduction

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