LMFDB - Embedded newform 8450.2.a.cs.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8450,2,Mod(1,8450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8450.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,8,0,8,0,0,10,8,16,0,0,0,0,10,0,8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.4735897080$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 20x^{6} + 132x^{4} - 332x^{2} + 256 $ x^8 - 20x^6 + 132x^4 - 332*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.83766$ of defining polynomial
Character	$\chi$	$=$	8450.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.83766 q^{3} +1.00000 q^{4} -1.83766 q^{6} +4.79742 q^{7} +1.00000 q^{8} +0.376989 q^{9} +1.65153 q^{11} -1.83766 q^{12} +4.79742 q^{14} +1.00000 q^{16} -0.0805241 q^{17} +0.376989 q^{18} -4.24776 q^{19} -8.81603 q^{21} +1.65153 q^{22} -5.89928 q^{23} -1.83766 q^{24} +4.82020 q^{27} +4.79742 q^{28} +4.42044 q^{29} +4.06163 q^{31} +1.00000 q^{32} -3.03494 q^{33} -0.0805241 q^{34} +0.376989 q^{36} +1.81708 q^{37} -4.24776 q^{38} +6.78855 q^{41} -8.81603 q^{42} -8.49552 q^{43} +1.65153 q^{44} -5.89928 q^{46} -0.448597 q^{47} -1.83766 q^{48} +16.0153 q^{49} +0.147976 q^{51} +11.5770 q^{53} +4.82020 q^{54} +4.79742 q^{56} +7.80593 q^{57} +4.42044 q^{58} +2.10800 q^{59} -3.12832 q^{61} +4.06163 q^{62} +1.80858 q^{63} +1.00000 q^{64} -3.03494 q^{66} +4.00000 q^{67} -0.0805241 q^{68} +10.8409 q^{69} -7.35063 q^{71} +0.376989 q^{72} -10.5752 q^{73} +1.81708 q^{74} -4.24776 q^{76} +7.92308 q^{77} +14.6468 q^{79} -9.98885 q^{81} +6.78855 q^{82} +12.4289 q^{83} -8.81603 q^{84} -8.49552 q^{86} -8.12325 q^{87} +1.65153 q^{88} -8.35955 q^{89} -5.89928 q^{92} -7.46388 q^{93} -0.448597 q^{94} -1.83766 q^{96} +5.80593 q^{97} +16.0153 q^{98} +0.622608 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} + 10 q^{7} + 8 q^{8} + 16 q^{9} + 10 q^{14} + 8 q^{16} + 16 q^{18} + 10 q^{28} - 6 q^{29} + 8 q^{32} + 20 q^{33} + 16 q^{36} + 40 q^{37} - 6 q^{47} + 30 q^{49} + 20 q^{51} + 10 q^{56}+ \cdots + 30 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 + 10 * q^7 + 8 * q^8 + 16 * q^9 + 10 * q^14 + 8 * q^16 + 16 * q^18 + 10 * q^28 - 6 * q^29 + 8 * q^32 + 20 * q^33 + 16 * q^36 + 40 * q^37 - 6 * q^47 + 30 * q^49 + 20 * q^51 + 10 * q^56 + 24 * q^57 - 6 * q^58 + 10 * q^61 + 50 * q^63 + 8 * q^64 + 20 * q^66 + 32 * q^67 + 4 * q^69 + 16 * q^72 + 26 * q^73 + 40 * q^74 - 4 * q^79 - 16 * q^81 + 48 * q^83 + 36 * q^93 - 6 * q^94 + 8 * q^97 + 30 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.83766	−1.06097	−0.530486	−	0.847693i	$-0.677991\pi$
$3$	−1.83766	−1.06097	−0.530486	+	0.847693i	$0.677991\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.83766	−0.750221
$7$	4.79742	1.81326	0.906628	−	0.421931i	$-0.138647\pi$
$7$	4.79742	1.81326	0.906628	+	0.421931i	$0.138647\pi$
$8$	1.00000	0.353553
$9$	0.376989	0.125663
$10$	0	0
$11$	1.65153	0.497954	0.248977	−	0.968509i	$-0.419906\pi$
$11$	1.65153	0.497954	0.248977	+	0.968509i	$0.419906\pi$
$12$	−1.83766	−0.530486
$13$	0	0
$13$	0	0
$14$	4.79742	1.28217
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.0805241	−0.0195300	−0.00976499	−	0.999952i	$-0.503108\pi$
$17$	−0.0805241	−0.0195300	−0.00976499	+	0.999952i	$0.503108\pi$
$18$	0.376989	0.0888572
$19$	−4.24776	−0.974502	−0.487251	−	0.873262i	$-0.662000\pi$
$19$	−4.24776	−0.974502	−0.487251	+	0.873262i	$0.662000\pi$
$20$	0	0
$21$	−8.81603	−1.92382
$22$	1.65153	0.352107
$23$	−5.89928	−1.23009	−0.615043	−	0.788494i	$-0.710861\pi$
$23$	−5.89928	−1.23009	−0.615043	+	0.788494i	$0.710861\pi$
$24$	−1.83766	−0.375111
$25$	0	0
$26$	0	0
$27$	4.82020	0.927648
$28$	4.79742	0.906628
$29$	4.42044	0.820854	0.410427	−	0.911893i	$-0.365380\pi$
$29$	4.42044	0.820854	0.410427	+	0.911893i	$0.365380\pi$
$30$	0	0
$31$	4.06163	0.729490	0.364745	−	0.931108i	$-0.381156\pi$
$31$	4.06163	0.729490	0.364745	+	0.931108i	$0.381156\pi$
$32$	1.00000	0.176777
$33$	−3.03494	−0.528316
$34$	−0.0805241	−0.0138098
$35$	0	0
$36$	0.376989	0.0628316
$37$	1.81708	0.298726	0.149363	−	0.988782i	$-0.452278\pi$
$37$	1.81708	0.298726	0.149363	+	0.988782i	$0.452278\pi$
$38$	−4.24776	−0.689077
$39$	0	0
$40$	0	0
$41$	6.78855	1.06019	0.530097	−	0.847937i	$-0.322156\pi$
$41$	6.78855	1.06019	0.530097	+	0.847937i	$0.322156\pi$
$42$	−8.81603	−1.36034
$43$	−8.49552	−1.29555	−0.647777	−	0.761830i	$-0.724301\pi$
$43$	−8.49552	−1.29555	−0.647777	+	0.761830i	$0.724301\pi$
$44$	1.65153	0.248977
$45$	0	0
$46$	−5.89928	−0.869802
$47$	−0.448597	−0.0654345	−0.0327173	−	0.999465i	$-0.510416\pi$
$47$	−0.448597	−0.0654345	−0.0327173	+	0.999465i	$0.510416\pi$
$48$	−1.83766	−0.265243
$49$	16.0153	2.28790
$50$	0	0
$51$	0.147976	0.0207208
$52$	0	0
$53$	11.5770	1.59022	0.795112	−	0.606463i	$-0.207412\pi$
$53$	11.5770	1.59022	0.795112	+	0.606463i	$0.207412\pi$
$54$	4.82020	0.655946
$55$	0	0
$56$	4.79742	0.641083
$57$	7.80593	1.03392
$58$	4.42044	0.580432
$59$	2.10800	0.274439	0.137219	−	0.990541i	$-0.456183\pi$
$59$	2.10800	0.274439	0.137219	+	0.990541i	$0.456183\pi$
$60$	0	0
$61$	−3.12832	−0.400540	−0.200270	−	0.979741i	$-0.564182\pi$
$61$	−3.12832	−0.400540	−0.200270	+	0.979741i	$0.564182\pi$
$62$	4.06163	0.515827
$63$	1.80858	0.227859
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.03494	−0.373576
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−0.0805241	−0.00976499
$69$	10.8409	1.30509
$70$	0	0
$71$	−7.35063	−0.872360	−0.436180	−	0.899859i	$-0.643669\pi$
$71$	−7.35063	−0.872360	−0.436180	+	0.899859i	$0.643669\pi$
$72$	0.376989	0.0444286
$73$	−10.5752	−1.23773	−0.618866	−	0.785496i	$-0.712408\pi$
$73$	−10.5752	−1.23773	−0.618866	+	0.785496i	$0.712408\pi$
$74$	1.81708	0.211231
$75$	0	0
$76$	−4.24776	−0.487251
$77$	7.92308	0.902918
$78$	0	0
$79$	14.6468	1.64789	0.823947	−	0.566667i	$-0.191767\pi$
$79$	14.6468	1.64789	0.823947	+	0.566667i	$0.191767\pi$
$80$	0	0
$81$	−9.98885	−1.10987
$82$	6.78855	0.749670
$83$	12.4289	1.36425	0.682127	−	0.731234i	$-0.261055\pi$
$83$	12.4289	1.36425	0.682127	+	0.731234i	$0.261055\pi$
$84$	−8.81603	−0.961908
$85$	0	0
$86$	−8.49552	−0.916095
$87$	−8.12325	−0.870904
$88$	1.65153	0.176053
$89$	−8.35955	−0.886111	−0.443055	−	0.896494i	$-0.646105\pi$
$89$	−8.35955	−0.886111	−0.443055	+	0.896494i	$0.646105\pi$
$90$	0	0
$91$	0	0
$92$	−5.89928	−0.615043
$93$	−7.46388	−0.773968
$94$	−0.448597	−0.0462692
$95$	0	0
$96$	−1.83766	−0.187555
$97$	5.80593	0.589503	0.294751	−	0.955574i	$-0.404763\pi$
$97$	5.80593	0.589503	0.294751	+	0.955574i	$0.404763\pi$
$98$	16.0153	1.61779
$99$	0.622608	0.0625744
$100$	0	0
$101$	−1.23752	−0.123138	−0.0615688	−	0.998103i	$-0.519610\pi$
$101$	−1.23752	−0.123138	−0.0615688	+	0.998103i	$0.519610\pi$
$102$	0.147976	0.0146518
$103$	9.38847	0.925073	0.462537	−	0.886600i	$-0.346939\pi$
$103$	9.38847	0.925073	0.462537	+	0.886600i	$0.346939\pi$
$104$	0	0
$105$	0	0
$106$	11.5770	1.12446
$107$	18.5066	1.78910	0.894550	−	0.446968i	$-0.147496\pi$
$107$	18.5066	1.78910	0.894550	+	0.446968i	$0.147496\pi$
$108$	4.82020	0.463824
$109$	−9.08638	−0.870317	−0.435158	−	0.900354i	$-0.643308\pi$
$109$	−9.08638	−0.870317	−0.435158	+	0.900354i	$0.643308\pi$
$110$	0	0
$111$	−3.33918	−0.316941
$112$	4.79742	0.453314
$113$	15.9407	1.49958	0.749788	−	0.661678i	$-0.230155\pi$
$113$	15.9407	1.49958	0.749788	+	0.661678i	$0.230155\pi$
$114$	7.80593	0.731092
$115$	0	0
$116$	4.42044	0.410427
$117$	0	0
$118$	2.10800	0.194058
$119$	−0.386309	−0.0354128
$120$	0	0
$121$	−8.27246	−0.752042
$122$	−3.12832	−0.283225
$123$	−12.4750	−1.12484
$124$	4.06163	0.364745
$125$	0	0
$126$	1.80858	0.161121
$127$	−0.892954	−0.0792369	−0.0396184	−	0.999215i	$-0.512614\pi$
$127$	−0.892954	−0.0792369	−0.0396184	+	0.999215i	$0.512614\pi$
$128$	1.00000	0.0883883
$129$	15.6119	1.37455
$130$	0	0
$131$	−1.11770	−0.0976541	−0.0488271	−	0.998807i	$-0.515548\pi$
$131$	−1.11770	−0.0976541	−0.0488271	+	0.998807i	$0.515548\pi$
$132$	−3.03494	−0.264158
$133$	−20.3783	−1.76702
$134$	4.00000	0.345547
$135$	0	0
$136$	−0.0805241	−0.00690489
$137$	−8.20936	−0.701373	−0.350686	−	0.936493i	$-0.614052\pi$
$137$	−8.20936	−0.701373	−0.350686	+	0.936493i	$0.614052\pi$
$138$	10.8409	0.922836
$139$	5.11770	0.434078	0.217039	−	0.976163i	$-0.430360\pi$
$139$	5.11770	0.434078	0.217039	+	0.976163i	$0.430360\pi$
$140$	0	0
$141$	0.824367	0.0694242
$142$	−7.35063	−0.616852
$143$	0	0
$144$	0.376989	0.0314158
$145$	0	0
$146$	−10.5752	−0.875209
$147$	−29.4306	−2.42740
$148$	1.81708	0.149363
$149$	−4.14215	−0.339338	−0.169669	−	0.985501i	$-0.554270\pi$
$149$	−4.14215	−0.339338	−0.169669	+	0.985501i	$0.554270\pi$
$150$	0	0
$151$	−4.43389	−0.360825	−0.180412	−	0.983591i	$-0.557743\pi$
$151$	−4.43389	−0.360825	−0.180412	+	0.983591i	$0.557743\pi$
$152$	−4.24776	−0.344539
$153$	−0.0303567	−0.00245420
$154$	7.92308	0.638460
$155$	0	0
$156$	0	0
$157$	8.32411	0.664337	0.332168	−	0.943220i	$-0.392220\pi$
$157$	8.32411	0.664337	0.332168	+	0.943220i	$0.392220\pi$
$158$	14.6468	1.16524
$159$	−21.2746	−1.68718
$160$	0	0
$161$	−28.3014	−2.23046
$162$	−9.98885	−0.784798
$163$	0.943288	0.0738840	0.0369420	−	0.999317i	$-0.488238\pi$
$163$	0.943288	0.0738840	0.0369420	+	0.999317i	$0.488238\pi$
$164$	6.78855	0.530097
$165$	0	0
$166$	12.4289	0.964673
$167$	4.30062	0.332792	0.166396	−	0.986059i	$-0.446787\pi$
$167$	4.30062	0.332792	0.166396	+	0.986059i	$0.446787\pi$
$168$	−8.81603	−0.680171
$169$	0	0
$170$	0	0
$171$	−1.60136	−0.122459
$172$	−8.49552	−0.647777
$173$	0.994872	0.0756387	0.0378194	−	0.999285i	$-0.487959\pi$
$173$	0.994872	0.0756387	0.0378194	+	0.999285i	$0.487959\pi$
$174$	−8.12325	−0.615822
$175$	0	0
$176$	1.65153	0.124489
$177$	−3.87379	−0.291172
$178$	−8.35955	−0.626575
$179$	−20.4357	−1.52744	−0.763719	−	0.645549i	$-0.776629\pi$
$179$	−20.4357	−1.52744	−0.763719	+	0.645549i	$0.776629\pi$
$180$	0	0
$181$	19.9522	1.48303	0.741517	−	0.670934i	$-0.234107\pi$
$181$	19.9522	1.48303	0.741517	+	0.670934i	$0.234107\pi$
$182$	0	0
$183$	5.74878	0.424962
$184$	−5.89928	−0.434901
$185$	0	0
$186$	−7.46388	−0.547278
$187$	−0.132988	−0.00972503
$188$	−0.448597	−0.0327173
$189$	23.1245	1.68206
$190$	0	0
$191$	−7.67972	−0.555685	−0.277843	−	0.960627i	$-0.589619\pi$
$191$	−7.67972	−0.555685	−0.277843	+	0.960627i	$0.589619\pi$
$192$	−1.83766	−0.132622
$193$	15.1914	1.09350	0.546751	−	0.837295i	$-0.315864\pi$
$193$	15.1914	1.09350	0.546751	+	0.837295i	$0.315864\pi$
$194$	5.80593	0.416841
$195$	0	0
$196$	16.0153	1.14395
$197$	−3.01327	−0.214686	−0.107343	−	0.994222i	$-0.534234\pi$
$197$	−3.01327	−0.214686	−0.107343	+	0.994222i	$0.534234\pi$
$198$	0.622608	0.0442468
$199$	−9.72106	−0.689107	−0.344554	−	0.938767i	$-0.611970\pi$
$199$	−9.72106	−0.689107	−0.344554	+	0.938767i	$0.611970\pi$
$200$	0	0
$201$	−7.35063	−0.518474
$202$	−1.23752	−0.0870714
$203$	21.2067	1.48842
$204$	0.147976	0.0103604
$205$	0	0
$206$	9.38847	0.654126
$207$	−2.22397	−0.154576
$208$	0	0
$209$	−7.01529	−0.485257
$210$	0	0
$211$	−4.79064	−0.329802	−0.164901	−	0.986310i	$-0.552730\pi$
$211$	−4.79064	−0.329802	−0.164901	+	0.986310i	$0.552730\pi$
$212$	11.5770	0.795112
$213$	13.5080	0.925550
$214$	18.5066	1.26508
$215$	0	0
$216$	4.82020	0.327973
$217$	19.4853	1.32275
$218$	−9.08638	−0.615407
$219$	19.4336	1.31320
$220$	0	0
$221$	0	0
$222$	−3.33918	−0.224111
$223$	14.0434	0.940419	0.470209	−	0.882555i	$-0.344178\pi$
$223$	14.0434	0.940419	0.470209	+	0.882555i	$0.344178\pi$
$224$	4.79742	0.320541
$225$	0	0
$226$	15.9407	1.06036
$227$	−2.88019	−0.191165	−0.0955823	−	0.995422i	$-0.530471\pi$
$227$	−2.88019	−0.191165	−0.0955823	+	0.995422i	$0.530471\pi$
$228$	7.80593	0.516960
$229$	−6.70802	−0.443279	−0.221639	−	0.975129i	$-0.571141\pi$
$229$	−6.70802	−0.443279	−0.221639	+	0.975129i	$0.571141\pi$
$230$	0	0
$231$	−14.5599	−0.957972
$232$	4.42044	0.290216
$233$	−17.5959	−1.15275	−0.576374	−	0.817186i	$-0.695533\pi$
$233$	−17.5959	−1.15275	−0.576374	+	0.817186i	$0.695533\pi$
$234$	0	0
$235$	0	0
$236$	2.10800	0.137219
$237$	−26.9158	−1.74837
$238$	−0.386309	−0.0250407
$239$	−11.3119	−0.731708	−0.365854	−	0.930672i	$-0.619223\pi$
$239$	−11.3119	−0.731708	−0.365854	+	0.930672i	$0.619223\pi$
$240$	0	0
$241$	5.08019	0.327244	0.163622	−	0.986523i	$-0.447682\pi$
$241$	5.08019	0.327244	0.163622	+	0.986523i	$0.447682\pi$
$242$	−8.27246	−0.531774
$243$	3.89550	0.249896
$244$	−3.12832	−0.200270
$245$	0	0
$246$	−12.4750	−0.795379
$247$	0	0
$248$	4.06163	0.257913
$249$	−22.8401	−1.44744
$250$	0	0
$251$	19.5704	1.23527	0.617637	−	0.786463i	$-0.288090\pi$
$251$	19.5704	1.23527	0.617637	+	0.786463i	$0.288090\pi$
$252$	1.80858	0.113930
$253$	−9.74283	−0.612526
$254$	−0.892954	−0.0560289
$255$	0	0
$256$	1.00000	0.0625000
$257$	−1.98398	−0.123757	−0.0618786	−	0.998084i	$-0.519709\pi$
$257$	−1.98398	−0.123757	−0.0618786	+	0.998084i	$0.519709\pi$
$258$	15.6119	0.971952
$259$	8.71731	0.541668
$260$	0	0
$261$	1.66646	0.103151
$262$	−1.11770	−0.0690519
$263$	0.608563	0.0375256	0.0187628	−	0.999824i	$-0.494027\pi$
$263$	0.608563	0.0375256	0.0187628	+	0.999824i	$0.494027\pi$
$264$	−3.03494	−0.186788
$265$	0	0
$266$	−20.3783	−1.24947
$267$	15.3620	0.940139
$268$	4.00000	0.244339
$269$	−17.7448	−1.08192	−0.540961	−	0.841048i	$-0.681939\pi$
$269$	−17.7448	−1.08192	−0.540961	+	0.841048i	$0.681939\pi$
$270$	0	0
$271$	−16.0279	−0.973626	−0.486813	−	0.873506i	$-0.661841\pi$
$271$	−16.0279	−0.973626	−0.486813	+	0.873506i	$0.661841\pi$
$272$	−0.0805241	−0.00488249
$273$	0	0
$274$	−8.20936	−0.495945
$275$	0	0
$276$	10.8409	0.652544
$277$	11.9042	0.715253	0.357626	−	0.933865i	$-0.383586\pi$
$277$	11.9042	0.715253	0.357626	+	0.933865i	$0.383586\pi$
$278$	5.11770	0.306939
$279$	1.53119	0.0916699
$280$	0	0
$281$	11.3975	0.679920	0.339960	−	0.940440i	$-0.389587\pi$
$281$	11.3975	0.679920	0.339960	+	0.940440i	$0.389587\pi$
$282$	0.824367	0.0490904
$283$	−20.4123	−1.21339	−0.606694	−	0.794935i	$-0.707505\pi$
$283$	−20.4123	−1.21339	−0.606694	+	0.794935i	$0.707505\pi$
$284$	−7.35063	−0.436180
$285$	0	0
$286$	0	0
$287$	32.5676	1.92240
$288$	0.376989	0.0222143
$289$	−16.9935	−0.999619
$290$	0	0
$291$	−10.6693	−0.625446
$292$	−10.5752	−0.618866
$293$	4.18768	0.244647	0.122323	−	0.992490i	$-0.460965\pi$
$293$	4.18768	0.244647	0.122323	+	0.992490i	$0.460965\pi$
$294$	−29.4306	−1.71643
$295$	0	0
$296$	1.81708	0.105616
$297$	7.96069	0.461926
$298$	−4.14215	−0.239948
$299$	0	0
$300$	0	0
$301$	−40.7566	−2.34917
$302$	−4.43389	−0.255142
$303$	2.27413	0.130646
$304$	−4.24776	−0.243626
$305$	0	0
$306$	−0.0303567	−0.00173538
$307$	18.0170	1.02828	0.514142	−	0.857705i	$-0.328110\pi$
$307$	18.0170	1.02828	0.514142	+	0.857705i	$0.328110\pi$
$308$	7.92308	0.451459
$309$	−17.2528	−0.981478
$310$	0	0
$311$	5.17816	0.293626	0.146813	−	0.989164i	$-0.453098\pi$
$311$	5.17816	0.293626	0.146813	+	0.989164i	$0.453098\pi$
$312$	0	0
$313$	32.6333	1.84455	0.922273	−	0.386539i	$-0.126330\pi$
$313$	32.6333	1.84455	0.922273	+	0.386539i	$0.126330\pi$
$314$	8.32411	0.469757
$315$	0	0
$316$	14.6468	0.823947
$317$	23.7385	1.33328	0.666642	−	0.745378i	$-0.267731\pi$
$317$	23.7385	1.33328	0.666642	+	0.745378i	$0.267731\pi$
$318$	−21.2746	−1.19302
$319$	7.30047	0.408748
$320$	0	0
$321$	−34.0088	−1.89819
$322$	−28.3014	−1.57717
$323$	0.342047	0.0190320
$324$	−9.98885	−0.554936
$325$	0	0
$326$	0.943288	0.0522439
$327$	16.6977	0.923383
$328$	6.78855	0.374835
$329$	−2.15211	−0.118650
$330$	0	0
$331$	30.4752	1.67507	0.837534	−	0.546386i	$-0.183997\pi$
$331$	30.4752	1.67507	0.837534	+	0.546386i	$0.183997\pi$
$332$	12.4289	0.682127
$333$	0.685020	0.0375389
$334$	4.30062	0.235319
$335$	0	0
$336$	−8.81603	−0.480954
$337$	19.2799	1.05024	0.525121	−	0.851027i	$-0.324020\pi$
$337$	19.2799	1.05024	0.525121	+	0.851027i	$0.324020\pi$
$338$	0	0
$339$	−29.2936	−1.59101
$340$	0	0
$341$	6.70788	0.363252
$342$	−1.60136	−0.0865916
$343$	43.2502	2.33529
$344$	−8.49552	−0.458048
$345$	0	0
$346$	0.994872	0.0534846
$347$	15.3720	0.825211	0.412605	−	0.910910i	$-0.364619\pi$
$347$	15.3720	0.825211	0.412605	+	0.910910i	$0.364619\pi$
$348$	−8.12325	−0.435452
$349$	15.2035	0.813827	0.406913	−	0.913467i	$-0.366605\pi$
$349$	15.2035	0.813827	0.406913	+	0.913467i	$0.366605\pi$
$350$	0	0
$351$	0	0
$352$	1.65153	0.0880267
$353$	−26.9017	−1.43183	−0.715917	−	0.698185i	$-0.753991\pi$
$353$	−26.9017	−1.43183	−0.715917	+	0.698185i	$0.753991\pi$
$354$	−3.87379	−0.205890
$355$	0	0
$356$	−8.35955	−0.443055
$357$	0.709903	0.0375721
$358$	−20.4357	−1.08006
$359$	−20.8348	−1.09962	−0.549809	−	0.835291i	$-0.685299\pi$
$359$	−20.8348	−1.09962	−0.549809	+	0.835291i	$0.685299\pi$
$360$	0	0
$361$	−0.956554	−0.0503449
$362$	19.9522	1.04866
$363$	15.2020	0.797896
$364$	0	0
$365$	0	0
$366$	5.74878	0.300494
$367$	34.5729	1.80469	0.902346	−	0.431013i	$-0.141844\pi$
$367$	34.5729	1.80469	0.902346	+	0.431013i	$0.141844\pi$
$368$	−5.89928	−0.307521
$369$	2.55921	0.133227
$370$	0	0
$371$	55.5398	2.88348
$372$	−7.46388	−0.386984
$373$	25.0064	1.29478	0.647391	−	0.762158i	$-0.275860\pi$
$373$	25.0064	1.29478	0.647391	+	0.762158i	$0.275860\pi$
$374$	−0.132988	−0.00687663
$375$	0	0
$376$	−0.448597	−0.0231346
$377$	0	0
$378$	23.1245	1.18940
$379$	−3.75953	−0.193114	−0.0965571	−	0.995327i	$-0.530783\pi$
$379$	−3.75953	−0.193114	−0.0965571	+	0.995327i	$0.530783\pi$
$380$	0	0
$381$	1.64094	0.0840682
$382$	−7.67972	−0.392929
$383$	3.93892	0.201269	0.100635	−	0.994923i	$-0.467913\pi$
$383$	3.93892	0.201269	0.100635	+	0.994923i	$0.467913\pi$
$384$	−1.83766	−0.0937776
$385$	0	0
$386$	15.1914	0.773223
$387$	−3.20272	−0.162803
$388$	5.80593	0.294751
$389$	−19.1589	−0.971394	−0.485697	−	0.874127i	$-0.661434\pi$
$389$	−19.1589	−0.971394	−0.485697	+	0.874127i	$0.661434\pi$
$390$	0	0
$391$	0.475035	0.0240235
$392$	16.0153	0.808894
$393$	2.05396	0.103608
$394$	−3.01327	−0.151806
$395$	0	0
$396$	0.622608	0.0312872
$397$	−6.45538	−0.323986	−0.161993	−	0.986792i	$-0.551792\pi$
$397$	−6.45538	−0.323986	−0.161993	+	0.986792i	$0.551792\pi$
$398$	−9.72106	−0.487273
$399$	37.4484	1.87476
$400$	0	0
$401$	34.6424	1.72996	0.864980	−	0.501807i	$-0.167331\pi$
$401$	34.6424	1.72996	0.864980	+	0.501807i	$0.167331\pi$
$402$	−7.35063	−0.366616
$403$	0	0
$404$	−1.23752	−0.0615688
$405$	0	0
$406$	21.2067	1.05247
$407$	3.00096	0.148752
$408$	0.147976	0.00732590
$409$	18.6071	0.920063	0.460031	−	0.887903i	$-0.347838\pi$
$409$	18.6071	0.920063	0.460031	+	0.887903i	$0.347838\pi$
$410$	0	0
$411$	15.0860	0.744137
$412$	9.38847	0.462537
$413$	10.1130	0.497628
$414$	−2.22397	−0.109302
$415$	0	0
$416$	0	0
$417$	−9.40459	−0.460545
$418$	−7.01529	−0.343129
$419$	22.9495	1.12116	0.560579	−	0.828101i	$-0.310579\pi$
$419$	22.9495	1.12116	0.560579	+	0.828101i	$0.310579\pi$
$420$	0	0
$421$	34.8196	1.69700	0.848501	−	0.529194i	$-0.177505\pi$
$421$	34.8196	1.69700	0.848501	+	0.529194i	$0.177505\pi$
$422$	−4.79064	−0.233205
$423$	−0.169116	−0.00822271
$424$	11.5770	0.562229
$425$	0	0
$426$	13.5080	0.654463
$427$	−15.0079	−0.726282
$428$	18.5066	0.894550
$429$	0	0
$430$	0	0
$431$	−20.4986	−0.987385	−0.493692	−	0.869637i	$-0.664353\pi$
$431$	−20.4986	−0.987385	−0.493692	+	0.869637i	$0.664353\pi$
$432$	4.82020	0.231912
$433$	2.14503	0.103083	0.0515417	−	0.998671i	$-0.483587\pi$
$433$	2.14503	0.103083	0.0515417	+	0.998671i	$0.483587\pi$
$434$	19.4853	0.935326
$435$	0	0
$436$	−9.08638	−0.435158
$437$	25.0587	1.19872
$438$	19.4336	0.928573
$439$	1.05195	0.0502068	0.0251034	−	0.999685i	$-0.492008\pi$
$439$	1.05195	0.0502068	0.0251034	+	0.999685i	$0.492008\pi$
$440$	0	0
$441$	6.03759	0.287504
$442$	0	0
$443$	18.3043	0.869665	0.434833	−	0.900511i	$-0.356807\pi$
$443$	18.3043	0.869665	0.434833	+	0.900511i	$0.356807\pi$
$444$	−3.33918	−0.158470
$445$	0	0
$446$	14.0434	0.664976
$447$	7.61186	0.360029
$448$	4.79742	0.226657
$449$	−13.9524	−0.658453	−0.329226	−	0.944251i	$-0.606788\pi$
$449$	−13.9524	−0.658453	−0.329226	+	0.944251i	$0.606788\pi$
$450$	0	0
$451$	11.2115	0.527927
$452$	15.9407	0.749788
$453$	8.14798	0.382825
$454$	−2.88019	−0.135174
$455$	0	0
$456$	7.80593	0.365546
$457$	−29.4331	−1.37682	−0.688411	−	0.725321i	$-0.741691\pi$
$457$	−29.4331	−1.37682	−0.688411	+	0.725321i	$0.741691\pi$
$458$	−6.70802	−0.313445
$459$	−0.388142	−0.0181169
$460$	0	0
$461$	9.09372	0.423537	0.211768	−	0.977320i	$-0.432078\pi$
$461$	9.09372	0.423537	0.211768	+	0.977320i	$0.432078\pi$
$462$	−14.5599	−0.677388
$463$	41.6642	1.93630	0.968150	−	0.250372i	$-0.0805529\pi$
$463$	41.6642	1.93630	0.968150	+	0.250372i	$0.0805529\pi$
$464$	4.42044	0.205214
$465$	0	0
$466$	−17.5959	−0.815116
$467$	−38.1079	−1.76342	−0.881712	−	0.471789i	$-0.843609\pi$
$467$	−38.1079	−1.76342	−0.881712	+	0.471789i	$0.843609\pi$
$468$	0	0
$469$	19.1897	0.886098
$470$	0	0
$471$	−15.2969	−0.704843
$472$	2.10800	0.0970288
$473$	−14.0306	−0.645126
$474$	−26.9158	−1.23628
$475$	0	0
$476$	−0.386309	−0.0177064
$477$	4.36440	0.199832
$478$	−11.3119	−0.517395
$479$	15.0374	0.687077	0.343538	−	0.939139i	$-0.388374\pi$
$479$	15.0374	0.687077	0.343538	+	0.939139i	$0.388374\pi$
$480$	0	0
$481$	0	0
$482$	5.08019	0.231396
$483$	52.0083	2.36646
$484$	−8.27246	−0.376021
$485$	0	0
$486$	3.89550	0.176703
$487$	4.56256	0.206749	0.103375	−	0.994642i	$-0.467036\pi$
$487$	4.56256	0.206749	0.103375	+	0.994642i	$0.467036\pi$
$488$	−3.12832	−0.141612
$489$	−1.73344	−0.0783890
$490$	0	0
$491$	8.41606	0.379812	0.189906	−	0.981802i	$-0.439182\pi$
$491$	8.41606	0.379812	0.189906	+	0.981802i	$0.439182\pi$
$492$	−12.4750	−0.562418
$493$	−0.355952	−0.0160313
$494$	0	0
$495$	0	0
$496$	4.06163	0.182372
$497$	−35.2641	−1.58181
$498$	−22.8401	−1.02349
$499$	12.5587	0.562206	0.281103	−	0.959678i	$-0.409300\pi$
$499$	12.5587	0.562206	0.281103	+	0.959678i	$0.409300\pi$
$500$	0	0
$501$	−7.90307	−0.353083
$502$	19.5704	0.873471
$503$	−7.84324	−0.349713	−0.174856	−	0.984594i	$-0.555946\pi$
$503$	−7.84324	−0.349713	−0.174856	+	0.984594i	$0.555946\pi$
$504$	1.80858	0.0805605
$505$	0	0
$506$	−9.74283	−0.433121
$507$	0	0
$508$	−0.892954	−0.0396184
$509$	36.2850	1.60830	0.804152	−	0.594424i	$-0.202620\pi$
$509$	36.2850	1.60830	0.804152	+	0.594424i	$0.202620\pi$
$510$	0	0
$511$	−50.7337	−2.24433
$512$	1.00000	0.0441942
$513$	−20.4750	−0.903995
$514$	−1.98398	−0.0875095
$515$	0	0
$516$	15.6119	0.687274
$517$	−0.740869	−0.0325834
$518$	8.71731	0.383017
$519$	−1.82823	−0.0802506
$520$	0	0
$521$	1.95007	0.0854341	0.0427171	−	0.999087i	$-0.486399\pi$
$521$	1.95007	0.0854341	0.0427171	+	0.999087i	$0.486399\pi$
$522$	1.66646	0.0729388
$523$	−9.08638	−0.397319	−0.198660	−	0.980069i	$-0.563659\pi$
$523$	−9.08638	−0.397319	−0.198660	+	0.980069i	$0.563659\pi$
$524$	−1.11770	−0.0488271
$525$	0	0
$526$	0.608563	0.0265346
$527$	−0.327059	−0.0142469
$528$	−3.03494	−0.132079
$529$	11.8016	0.513111
$530$	0	0
$531$	0.794695	0.0344868
$532$	−20.3783	−0.883511
$533$	0	0
$534$	15.3620	0.664779
$535$	0	0
$536$	4.00000	0.172774
$537$	37.5539	1.62057
$538$	−17.7448	−0.765035
$539$	26.4497	1.13927
$540$	0	0
$541$	35.7827	1.53842	0.769209	−	0.638997i	$-0.220650\pi$
$541$	35.7827	1.53842	0.769209	+	0.638997i	$0.220650\pi$
$542$	−16.0279	−0.688458
$543$	−36.6653	−1.57346
$544$	−0.0805241	−0.00345244
$545$	0	0
$546$	0	0
$547$	−29.0584	−1.24245	−0.621223	−	0.783634i	$-0.713364\pi$
$547$	−29.0584	−1.24245	−0.621223	+	0.783634i	$0.713364\pi$
$548$	−8.20936	−0.350686
$549$	−1.17934	−0.0503331
$550$	0	0
$551$	−18.7769	−0.799925
$552$	10.8409	0.461418
$553$	70.2669	2.98805
$554$	11.9042	0.505760
$555$	0	0
$556$	5.11770	0.217039
$557$	−33.0984	−1.40243	−0.701213	−	0.712952i	$-0.747358\pi$
$557$	−33.0984	−1.40243	−0.701213	+	0.712952i	$0.747358\pi$
$558$	1.53119	0.0648204
$559$	0	0
$560$	0	0
$561$	0.244386	0.0103180
$562$	11.3975	0.480776
$563$	10.4632	0.440971	0.220485	−	0.975390i	$-0.429236\pi$
$563$	10.4632	0.440971	0.220485	+	0.975390i	$0.429236\pi$
$564$	0.824367	0.0347121
$565$	0	0
$566$	−20.4123	−0.857995
$567$	−47.9207	−2.01248
$568$	−7.35063	−0.308426
$569$	8.63244	0.361891	0.180945	−	0.983493i	$-0.442084\pi$
$569$	8.63244	0.361891	0.180945	+	0.983493i	$0.442084\pi$
$570$	0	0
$571$	−14.3069	−0.598724	−0.299362	−	0.954140i	$-0.596774\pi$
$571$	−14.3069	−0.598724	−0.299362	+	0.954140i	$0.596774\pi$
$572$	0	0
$573$	14.1127	0.589567
$574$	32.5676	1.35934
$575$	0	0
$576$	0.376989	0.0157079
$577$	11.9697	0.498306	0.249153	−	0.968464i	$-0.419848\pi$
$577$	11.9697	0.498306	0.249153	+	0.968464i	$0.419848\pi$
$578$	−16.9935	−0.706837
$579$	−27.9166	−1.16018
$580$	0	0
$581$	59.6269	2.47374
$582$	−10.6693	−0.442257
$583$	19.1197	0.791858
$584$	−10.5752	−0.437605
$585$	0	0
$586$	4.18768	0.172991
$587$	5.50796	0.227338	0.113669	−	0.993519i	$-0.463740\pi$
$587$	5.50796	0.227338	0.113669	+	0.993519i	$0.463740\pi$
$588$	−29.4306	−1.21370
$589$	−17.2528	−0.710889
$590$	0	0
$591$	5.53735	0.227776
$592$	1.81708	0.0746816
$593$	−24.6037	−1.01035	−0.505177	−	0.863016i	$-0.668573\pi$
$593$	−24.6037	−1.01035	−0.505177	+	0.863016i	$0.668573\pi$
$594$	7.96069	0.326631
$595$	0	0
$596$	−4.14215	−0.169669
$597$	17.8640	0.731124
$598$	0	0
$599$	−35.3159	−1.44297	−0.721484	−	0.692431i	$-0.756540\pi$
$599$	−35.3159	−1.44297	−0.721484	+	0.692431i	$0.756540\pi$
$600$	0	0
$601$	34.9214	1.42447	0.712236	−	0.701940i	$-0.247682\pi$
$601$	34.9214	1.42447	0.712236	+	0.701940i	$0.247682\pi$
$602$	−40.7566	−1.66111
$603$	1.50796	0.0614088
$604$	−4.43389	−0.180412
$605$	0	0
$606$	2.27413	0.0923804
$607$	−24.9627	−1.01320	−0.506602	−	0.862180i	$-0.669099\pi$
$607$	−24.9627	−1.01320	−0.506602	+	0.862180i	$0.669099\pi$
$608$	−4.24776	−0.172269
$609$	−38.9707	−1.57917
$610$	0	0
$611$	0	0
$612$	−0.0303567	−0.00122710
$613$	16.1266	0.651347	0.325674	−	0.945482i	$-0.394409\pi$
$613$	16.1266	0.651347	0.325674	+	0.945482i	$0.394409\pi$
$614$	18.0170	0.727107
$615$	0	0
$616$	7.92308	0.319230
$617$	−28.5073	−1.14766	−0.573831	−	0.818974i	$-0.694543\pi$
$617$	−28.5073	−1.14766	−0.573831	+	0.818974i	$0.694543\pi$
$618$	−17.2528	−0.694010
$619$	0.338217	0.0135941	0.00679705	−	0.999977i	$-0.497836\pi$
$619$	0.338217	0.0135941	0.00679705	+	0.999977i	$0.497836\pi$
$620$	0	0
$621$	−28.4357	−1.14109
$622$	5.17816	0.207625
$623$	−40.1043	−1.60675
$624$	0	0
$625$	0	0
$626$	32.6333	1.30429
$627$	12.8917	0.514845
$628$	8.32411	0.332168
$629$	−0.146319	−0.00583412
$630$	0	0
$631$	−32.1930	−1.28158	−0.640791	−	0.767715i	$-0.721394\pi$
$631$	−32.1930	−1.28158	−0.640791	+	0.767715i	$0.721394\pi$
$632$	14.6468	0.582618
$633$	8.80357	0.349910
$634$	23.7385	0.942774
$635$	0	0
$636$	−21.2746	−0.843592
$637$	0	0
$638$	7.30047	0.289028
$639$	−2.77111	−0.109623
$640$	0	0
$641$	12.5996	0.497655	0.248827	−	0.968548i	$-0.419955\pi$
$641$	12.5996	0.497655	0.248827	+	0.968548i	$0.419955\pi$
$642$	−34.0088	−1.34222
$643$	19.4055	0.765280	0.382640	−	0.923898i	$-0.375015\pi$
$643$	19.4055	0.765280	0.382640	+	0.923898i	$0.375015\pi$
$644$	−28.3014	−1.11523
$645$	0	0
$646$	0.342047	0.0134577
$647$	−18.6943	−0.734949	−0.367475	−	0.930034i	$-0.619777\pi$
$647$	−18.6943	−0.734949	−0.367475	+	0.930034i	$0.619777\pi$
$648$	−9.98885	−0.392399
$649$	3.48143	0.136658
$650$	0	0
$651$	−35.8074	−1.40340
$652$	0.943288	0.0369420
$653$	−17.8699	−0.699305	−0.349652	−	0.936879i	$-0.613700\pi$
$653$	−17.8699	−0.699305	−0.349652	+	0.936879i	$0.613700\pi$
$654$	16.6977	0.652930
$655$	0	0
$656$	6.78855	0.265048
$657$	−3.98673	−0.155537
$658$	−2.15211	−0.0838979
$659$	−39.0893	−1.52270	−0.761352	−	0.648339i	$-0.775464\pi$
$659$	−39.0893	−1.52270	−0.761352	+	0.648339i	$0.775464\pi$
$660$	0	0
$661$	−40.3696	−1.57020	−0.785098	−	0.619372i	$-0.787387\pi$
$661$	−40.3696	−1.57020	−0.785098	+	0.619372i	$0.787387\pi$
$662$	30.4752	1.18445
$663$	0	0
$664$	12.4289	0.482336
$665$	0	0
$666$	0.685020	0.0265440
$667$	−26.0774	−1.00972
$668$	4.30062	0.166396
$669$	−25.8071	−0.997759
$670$	0	0
$671$	−5.16650	−0.199451
$672$	−8.81603	−0.340086
$673$	21.1833	0.816558	0.408279	−	0.912857i	$-0.366129\pi$
$673$	21.1833	0.816558	0.408279	+	0.912857i	$0.366129\pi$
$674$	19.2799	0.742634
$675$	0	0
$676$	0	0
$677$	2.48027	0.0953245	0.0476622	−	0.998864i	$-0.484823\pi$
$677$	2.48027	0.0953245	0.0476622	+	0.998864i	$0.484823\pi$
$678$	−29.2936	−1.12501
$679$	27.8535	1.06892
$680$	0	0
$681$	5.29280	0.202820
$682$	6.70788	0.256858
$683$	−12.8579	−0.491993	−0.245997	−	0.969271i	$-0.579115\pi$
$683$	−12.8579	−0.491993	−0.245997	+	0.969271i	$0.579115\pi$
$684$	−1.60136	−0.0612295
$685$	0	0
$686$	43.2502	1.65130
$687$	12.3271	0.470307
$688$	−8.49552	−0.323888
$689$	0	0
$690$	0	0
$691$	−1.12646	−0.0428526	−0.0214263	−	0.999770i	$-0.506821\pi$
$691$	−1.12646	−0.0428526	−0.0214263	+	0.999770i	$0.506821\pi$
$692$	0.994872	0.0378194
$693$	2.98691	0.113464
$694$	15.3720	0.583512
$695$	0	0
$696$	−8.12325	−0.307911
$697$	−0.546642	−0.0207055
$698$	15.2035	0.575462
$699$	32.3353	1.22303
$700$	0	0
$701$	−9.39602	−0.354883	−0.177441	−	0.984131i	$-0.556782\pi$
$701$	−9.39602	−0.354883	−0.177441	+	0.984131i	$0.556782\pi$
$702$	0	0
$703$	−7.71852	−0.291110
$704$	1.65153	0.0622443
$705$	0	0
$706$	−26.9017	−1.01246
$707$	−5.93690	−0.223280
$708$	−3.87379	−0.145586
$709$	17.8713	0.671172	0.335586	−	0.942010i	$-0.391066\pi$
$709$	17.8713	0.671172	0.335586	+	0.942010i	$0.391066\pi$
$710$	0	0
$711$	5.52169	0.207079
$712$	−8.35955	−0.313287
$713$	−23.9607	−0.897335
$714$	0.709903	0.0265675
$715$	0	0
$716$	−20.4357	−0.763719
$717$	20.7875	0.776322
$718$	−20.8348	−0.777547
$719$	−2.31296	−0.0862588	−0.0431294	−	0.999069i	$-0.513733\pi$
$719$	−2.31296	−0.0862588	−0.0431294	+	0.999069i	$0.513733\pi$
$720$	0	0
$721$	45.0405	1.67740
$722$	−0.956554	−0.0355992
$723$	−9.33566	−0.347197
$724$	19.9522	0.741517
$725$	0	0
$726$	15.2020	0.564198
$727$	−17.9866	−0.667087	−0.333543	−	0.942735i	$-0.608244\pi$
$727$	−17.9866	−0.667087	−0.333543	+	0.942735i	$0.608244\pi$
$728$	0	0
$729$	22.8079	0.844739
$730$	0	0
$731$	0.684094	0.0253021
$732$	5.74878	0.212481
$733$	45.2898	1.67282	0.836408	−	0.548108i	$-0.184652\pi$
$733$	45.2898	1.67282	0.836408	+	0.548108i	$0.184652\pi$
$734$	34.5729	1.27611
$735$	0	0
$736$	−5.89928	−0.217451
$737$	6.60611	0.243339
$738$	2.55921	0.0942058
$739$	−45.0044	−1.65551	−0.827756	−	0.561088i	$-0.810383\pi$
$739$	−45.0044	−1.65551	−0.827756	+	0.561088i	$0.810383\pi$
$740$	0	0
$741$	0	0
$742$	55.5398	2.03893
$743$	6.56669	0.240908	0.120454	−	0.992719i	$-0.461565\pi$
$743$	6.56669	0.240908	0.120454	+	0.992719i	$0.461565\pi$
$744$	−7.46388	−0.273639
$745$	0	0
$746$	25.0064	0.915549
$747$	4.68558	0.171436
$748$	−0.132988	−0.00486251
$749$	88.7840	3.24410
$750$	0	0
$751$	−27.3499	−0.998013	−0.499006	−	0.866598i	$-0.666302\pi$
$751$	−27.3499	−0.998013	−0.499006	+	0.866598i	$0.666302\pi$
$752$	−0.448597	−0.0163586
$753$	−35.9638	−1.31059
$754$	0	0
$755$	0	0
$756$	23.1245	0.841031
$757$	−39.0526	−1.41939	−0.709696	−	0.704509i	$-0.751167\pi$
$757$	−39.0526	−1.41939	−0.709696	+	0.704509i	$0.751167\pi$
$758$	−3.75953	−0.136552
$759$	17.9040	0.649874
$760$	0	0
$761$	28.4342	1.03074	0.515369	−	0.856968i	$-0.327655\pi$
$761$	28.4342	1.03074	0.515369	+	0.856968i	$0.327655\pi$
$762$	1.64094	0.0594452
$763$	−43.5912	−1.57811
$764$	−7.67972	−0.277843
$765$	0	0
$766$	3.93892	0.142319
$767$	0	0
$768$	−1.83766	−0.0663108
$769$	−29.2424	−1.05451	−0.527254	−	0.849708i	$-0.676778\pi$
$769$	−29.2424	−1.05451	−0.527254	+	0.849708i	$0.676778\pi$
$770$	0	0
$771$	3.64587	0.131303
$772$	15.1914	0.546751
$773$	−20.8212	−0.748887	−0.374444	−	0.927250i	$-0.622166\pi$
$773$	−20.8212	−0.748887	−0.374444	+	0.927250i	$0.622166\pi$
$774$	−3.20272	−0.115119
$775$	0	0
$776$	5.80593	0.208421
$777$	−16.0194	−0.574694
$778$	−19.1589	−0.686879
$779$	−28.8361	−1.03316
$780$	0	0
$781$	−12.1398	−0.434395
$782$	0.475035	0.0169872
$783$	21.3074	0.761463
$784$	16.0153	0.571974
$785$	0	0
$786$	2.05396	0.0732622
$787$	20.9569	0.747031	0.373516	−	0.927624i	$-0.378152\pi$
$787$	20.9569	0.747031	0.373516	+	0.927624i	$0.378152\pi$
$788$	−3.01327	−0.107343
$789$	−1.11833	−0.0398137
$790$	0	0
$791$	76.4744	2.71912
$792$	0.622608	0.0221234
$793$	0	0
$794$	−6.45538	−0.229093
$795$	0	0
$796$	−9.72106	−0.344554
$797$	18.5740	0.657924	0.328962	−	0.944343i	$-0.393301\pi$
$797$	18.5740	0.657924	0.328962	+	0.944343i	$0.393301\pi$
$798$	37.4484	1.32566
$799$	0.0361228	0.00127793
$800$	0	0
$801$	−3.15146	−0.111351
$802$	34.6424	1.22327
$803$	−17.4652	−0.616334
$804$	−7.35063	−0.259237
$805$	0	0
$806$	0	0
$807$	32.6090	1.14789
$808$	−1.23752	−0.0435357
$809$	28.1410	0.989383	0.494692	−	0.869069i	$-0.335281\pi$
$809$	28.1410	0.989383	0.494692	+	0.869069i	$0.335281\pi$
$810$	0	0
$811$	17.1410	0.601903	0.300952	−	0.953639i	$-0.402696\pi$
$811$	17.1410	0.601903	0.300952	+	0.953639i	$0.402696\pi$
$812$	21.2067	0.744210
$813$	29.4538	1.03299
$814$	3.00096	0.105184
$815$	0	0
$816$	0.147976	0.00518019
$817$	36.0869	1.26252
$818$	18.6071	0.650583
$819$	0	0
$820$	0	0
$821$	−13.3822	−0.467043	−0.233521	−	0.972352i	$-0.575025\pi$
$821$	−13.3822	−0.467043	−0.233521	+	0.972352i	$0.575025\pi$
$822$	15.0860	0.526184
$823$	−21.6095	−0.753259	−0.376629	−	0.926364i	$-0.622917\pi$
$823$	−21.6095	−0.753259	−0.376629	+	0.926364i	$0.622917\pi$
$824$	9.38847	0.327063
$825$	0	0
$826$	10.1130	0.351876
$827$	5.85827	0.203712	0.101856	−	0.994799i	$-0.467522\pi$
$827$	5.85827	0.203712	0.101856	+	0.994799i	$0.467522\pi$
$828$	−2.22397	−0.0772882
$829$	−21.2783	−0.739026	−0.369513	−	0.929225i	$-0.620476\pi$
$829$	−21.2783	−0.739026	−0.369513	+	0.929225i	$0.620476\pi$
$830$	0	0
$831$	−21.8758	−0.758864
$832$	0	0
$833$	−1.28962	−0.0446826
$834$	−9.40459	−0.325654
$835$	0	0
$836$	−7.01529	−0.242629
$837$	19.5778	0.676709
$838$	22.9495	0.792778
$839$	−19.5222	−0.673981	−0.336990	−	0.941508i	$-0.609409\pi$
$839$	−19.5222	−0.673981	−0.336990	+	0.941508i	$0.609409\pi$
$840$	0	0
$841$	−9.45975	−0.326198
$842$	34.8196	1.19996
$843$	−20.9448	−0.721376
$844$	−4.79064	−0.164901
$845$	0	0
$846$	−0.169116	−0.00581433
$847$	−39.6865	−1.36364
$848$	11.5770	0.397556
$849$	37.5109	1.28737
$850$	0	0
$851$	−10.7195	−0.367459
$852$	13.5080	0.462775
$853$	33.3923	1.14333	0.571665	−	0.820487i	$-0.306298\pi$
$853$	33.3923	1.14333	0.571665	+	0.820487i	$0.306298\pi$
$854$	−15.0079	−0.513559
$855$	0	0
$856$	18.5066	0.632542
$857$	22.7514	0.777172	0.388586	−	0.921412i	$-0.372964\pi$
$857$	22.7514	0.777172	0.388586	+	0.921412i	$0.372964\pi$
$858$	0	0
$859$	7.99391	0.272749	0.136374	−	0.990657i	$-0.456455\pi$
$859$	7.99391	0.272749	0.136374	+	0.990657i	$0.456455\pi$
$860$	0	0
$861$	−59.8480	−2.03962
$862$	−20.4986	−0.698186
$863$	11.7205	0.398971	0.199486	−	0.979901i	$-0.436073\pi$
$863$	11.7205	0.398971	0.199486	+	0.979901i	$0.436073\pi$
$864$	4.82020	0.163986
$865$	0	0
$866$	2.14503	0.0728909
$867$	31.2283	1.06057
$868$	19.4853	0.661376
$869$	24.1896	0.820575
$870$	0	0
$871$	0	0
$872$	−9.08638	−0.307704
$873$	2.18877	0.0740787
$874$	25.0587	0.847624
$875$	0	0
$876$	19.4336	0.656600
$877$	1.72808	0.0583530	0.0291765	−	0.999574i	$-0.490712\pi$
$877$	1.72808	0.0583530	0.0291765	+	0.999574i	$0.490712\pi$
$878$	1.05195	0.0355016
$879$	−7.69553	−0.259564
$880$	0	0
$881$	−33.5662	−1.13087	−0.565436	−	0.824792i	$-0.691292\pi$
$881$	−33.5662	−1.13087	−0.565436	+	0.824792i	$0.691292\pi$
$882$	6.03759	0.203296
$883$	18.2526	0.614249	0.307124	−	0.951669i	$-0.400633\pi$
$883$	18.2526	0.614249	0.307124	+	0.951669i	$0.400633\pi$
$884$	0	0
$885$	0	0
$886$	18.3043	0.614946
$887$	−23.4791	−0.788351	−0.394176	−	0.919035i	$-0.628970\pi$
$887$	−23.4791	−0.788351	−0.394176	+	0.919035i	$0.628970\pi$
$888$	−3.33918	−0.112055
$889$	−4.28388	−0.143677
$890$	0	0
$891$	−16.4968	−0.552665
$892$	14.0434	0.470209
$893$	1.90553	0.0637661
$894$	7.61186	0.254579
$895$	0	0
$896$	4.79742	0.160271
$897$	0	0
$898$	−13.9524	−0.465596
$899$	17.9542	0.598805
$900$	0	0
$901$	−0.932228	−0.0310570
$902$	11.2115	0.373301
$903$	74.8967	2.49241
$904$	15.9407	0.530180
$905$	0	0
$906$	8.14798	0.270698
$907$	26.6800	0.885895	0.442948	−	0.896547i	$-0.353933\pi$
$907$	26.6800	0.885895	0.442948	+	0.896547i	$0.353933\pi$
$908$	−2.88019	−0.0955823
$909$	−0.466531	−0.0154739
$910$	0	0
$911$	−33.1591	−1.09861	−0.549305	−	0.835622i	$-0.685108\pi$
$911$	−33.1591	−1.09861	−0.549305	+	0.835622i	$0.685108\pi$
$912$	7.80593	0.258480
$913$	20.5267	0.679335
$914$	−29.4331	−0.973559
$915$	0	0
$916$	−6.70802	−0.221639
$917$	−5.36209	−0.177072
$918$	−0.388142	−0.0128106
$919$	−38.5789	−1.27260	−0.636301	−	0.771441i	$-0.719536\pi$
$919$	−38.5789	−1.27260	−0.636301	+	0.771441i	$0.719536\pi$
$920$	0	0
$921$	−33.1091	−1.09098
$922$	9.09372	0.299486
$923$	0	0
$924$	−14.5599	−0.478986
$925$	0	0
$926$	41.6642	1.36917
$927$	3.53935	0.116248
$928$	4.42044	0.145108
$929$	−24.5973	−0.807011	−0.403505	−	0.914977i	$-0.632208\pi$
$929$	−24.5973	−0.807011	−0.403505	+	0.914977i	$0.632208\pi$
$930$	0	0
$931$	−68.0291	−2.22956
$932$	−17.5959	−0.576374
$933$	−9.51568	−0.311530
$934$	−38.1079	−1.24693
$935$	0	0
$936$	0	0
$937$	−25.6787	−0.838886	−0.419443	−	0.907782i	$-0.637775\pi$
$937$	−25.6787	−0.838886	−0.419443	+	0.907782i	$0.637775\pi$
$938$	19.1897	0.626566
$939$	−59.9690	−1.95701
$940$	0	0
$941$	−27.7630	−0.905047	−0.452524	−	0.891752i	$-0.649476\pi$
$941$	−27.7630	−0.905047	−0.452524	+	0.891752i	$0.649476\pi$
$942$	−15.2969	−0.498399
$943$	−40.0476	−1.30413
$944$	2.10800	0.0686097
$945$	0	0
$946$	−14.0306	−0.456173
$947$	−10.0291	−0.325902	−0.162951	−	0.986634i	$-0.552101\pi$
$947$	−10.0291	−0.325902	−0.162951	+	0.986634i	$0.552101\pi$
$948$	−26.9158	−0.874185
$949$	0	0
$950$	0	0
$951$	−43.6232	−1.41458
$952$	−0.386309	−0.0125203
$953$	−34.9592	−1.13244	−0.566220	−	0.824254i	$-0.691595\pi$
$953$	−34.9592	−1.13244	−0.566220	+	0.824254i	$0.691595\pi$
$954$	4.36440	0.141303
$955$	0	0
$956$	−11.3119	−0.365854
$957$	−13.4158	−0.433670
$958$	15.0374	0.485837
$959$	−39.3838	−1.27177
$960$	0	0
$961$	−14.5032	−0.467845
$962$	0	0
$963$	6.97679	0.224824
$964$	5.08019	0.163622
$965$	0	0
$966$	52.0083	1.67334
$967$	−8.61013	−0.276883	−0.138442	−	0.990371i	$-0.544209\pi$
$967$	−8.61013	−0.276883	−0.138442	+	0.990371i	$0.544209\pi$
$968$	−8.27246	−0.265887
$969$	−0.628566	−0.0201924
$970$	0	0
$971$	−28.4894	−0.914268	−0.457134	−	0.889398i	$-0.651124\pi$
$971$	−28.4894	−0.914268	−0.457134	+	0.889398i	$0.651124\pi$
$972$	3.89550	0.124948
$973$	24.5518	0.787094
$974$	4.56256	0.146194
$975$	0	0
$976$	−3.12832	−0.100135
$977$	−23.1128	−0.739444	−0.369722	−	0.929142i	$-0.620547\pi$
$977$	−23.1128	−0.739444	−0.369722	+	0.929142i	$0.620547\pi$
$978$	−1.73344	−0.0554294
$979$	−13.8060	−0.441242
$980$	0	0
$981$	−3.42547	−0.109367
$982$	8.41606	0.268567
$983$	28.0211	0.893736	0.446868	−	0.894600i	$-0.352539\pi$
$983$	28.0211	0.893736	0.446868	+	0.894600i	$0.352539\pi$
$984$	−12.4750	−0.397690
$985$	0	0
$986$	−0.355952	−0.0113358
$987$	3.95484	0.125884
$988$	0	0
$989$	50.1175	1.59364
$990$	0	0
$991$	24.8714	0.790067	0.395034	−	0.918667i	$-0.370733\pi$
$991$	24.8714	0.790067	0.395034	+	0.918667i	$0.370733\pi$
$992$	4.06163	0.128957
$993$	−56.0030	−1.77720
$994$	−35.2641	−1.11851
$995$	0	0
$996$	−22.8401	−0.723718
$997$	26.6892	0.845254	0.422627	−	0.906304i	$-0.361108\pi$
$997$	26.6892	0.845254	0.422627	+	0.906304i	$0.361108\pi$
$998$	12.5587	0.397539
$999$	8.75869	0.277113

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.cs.1.3		8
5.2	odd	4		1690.2.b.e.339.14		16
5.3	odd	4		1690.2.b.e.339.3		16
5.4	even	2		8450.2.a.cr.1.6		8
13.6	odd	12		650.2.m.e.101.3		16
13.11	odd	12		650.2.m.e.251.3		16
13.12	even	2		8450.2.a.cr.1.3		8
65.8	even	4		1690.2.c.f.1689.3		8
65.12	odd	4		1690.2.b.e.339.6		16
65.18	even	4		1690.2.c.e.1689.3		8
65.19	odd	12		650.2.m.e.101.6		16
65.24	odd	12		650.2.m.e.251.6		16
65.32	even	12		130.2.m.a.49.3	✓	8
65.37	even	12		130.2.m.b.69.2	yes	8
65.38	odd	4		1690.2.b.e.339.11		16
65.47	even	4		1690.2.c.e.1689.6		8
65.57	even	4		1690.2.c.f.1689.6		8
65.58	even	12		130.2.m.b.49.2	yes	8
65.63	even	12		130.2.m.a.69.3	yes	8
65.64	even	2	inner	8450.2.a.cs.1.6		8
195.32	odd	12		1170.2.bj.b.829.4		8
195.128	odd	12		1170.2.bj.b.199.4		8
195.167	odd	12		1170.2.bj.a.199.1		8
195.188	odd	12		1170.2.bj.a.829.1		8
260.63	odd	12		1040.2.df.c.849.2		8
260.123	odd	12		1040.2.df.a.49.3		8
260.167	odd	12		1040.2.df.a.849.3		8
260.227	odd	12		1040.2.df.c.49.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.m.a.49.3	✓	8	65.32	even	12
130.2.m.a.69.3	yes	8	65.63	even	12
130.2.m.b.49.2	yes	8	65.58	even	12
130.2.m.b.69.2	yes	8	65.37	even	12
650.2.m.e.101.3		16	13.6	odd	12
650.2.m.e.101.6		16	65.19	odd	12
650.2.m.e.251.3		16	13.11	odd	12
650.2.m.e.251.6		16	65.24	odd	12
1040.2.df.a.49.3		8	260.123	odd	12
1040.2.df.a.849.3		8	260.167	odd	12
1040.2.df.c.49.2		8	260.227	odd	12
1040.2.df.c.849.2		8	260.63	odd	12
1170.2.bj.a.199.1		8	195.167	odd	12
1170.2.bj.a.829.1		8	195.188	odd	12
1170.2.bj.b.199.4		8	195.128	odd	12
1170.2.bj.b.829.4		8	195.32	odd	12
1690.2.b.e.339.3		16	5.3	odd	4
1690.2.b.e.339.6		16	65.12	odd	4
1690.2.b.e.339.11		16	65.38	odd	4
1690.2.b.e.339.14		16	5.2	odd	4
1690.2.c.e.1689.3		8	65.18	even	4
1690.2.c.e.1689.6		8	65.47	even	4
1690.2.c.f.1689.3		8	65.8	even	4
1690.2.c.f.1689.6		8	65.57	even	4
8450.2.a.cr.1.3		8	13.12	even	2
8450.2.a.cr.1.6		8	5.4	even	2
8450.2.a.cs.1.3		8	1.1	even	1	trivial
8450.2.a.cs.1.6		8	65.64	even	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:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.83766 q^{3} +1.00000 q^{4} -1.83766 q^{6} +4.79742 q^{7} +1.00000 q^{8} +0.376989 q^{9} +1.65153 q^{11} -1.83766 q^{12} +4.79742 q^{14} +1.00000 q^{16} -0.0805241 q^{17} +0.376989 q^{18} -4.24776 q^{19} -8.81603 q^{21} +1.65153 q^{22} -5.89928 q^{23} -1.83766 q^{24} +4.82020 q^{27} +4.79742 q^{28} +4.42044 q^{29} +4.06163 q^{31} +1.00000 q^{32} -3.03494 q^{33} -0.0805241 q^{34} +0.376989 q^{36} +1.81708 q^{37} -4.24776 q^{38} +6.78855 q^{41} -8.81603 q^{42} -8.49552 q^{43} +1.65153 q^{44} -5.89928 q^{46} -0.448597 q^{47} -1.83766 q^{48} +16.0153 q^{49} +0.147976 q^{51} +11.5770 q^{53} +4.82020 q^{54} +4.79742 q^{56} +7.80593 q^{57} +4.42044 q^{58} +2.10800 q^{59} -3.12832 q^{61} +4.06163 q^{62} +1.80858 q^{63} +1.00000 q^{64} -3.03494 q^{66} +4.00000 q^{67} -0.0805241 q^{68} +10.8409 q^{69} -7.35063 q^{71} +0.376989 q^{72} -10.5752 q^{73} +1.81708 q^{74} -4.24776 q^{76} +7.92308 q^{77} +14.6468 q^{79} -9.98885 q^{81} +6.78855 q^{82} +12.4289 q^{83} -8.81603 q^{84} -8.49552 q^{86} -8.12325 q^{87} +1.65153 q^{88} -8.35955 q^{89} -5.89928 q^{92} -7.46388 q^{93} -0.448597 q^{94} -1.83766 q^{96} +5.80593 q^{97} +16.0153 q^{98} +0.622608 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} + 10 q^{7} + 8 q^{8} + 16 q^{9} + 10 q^{14} + 8 q^{16} + 16 q^{18} + 10 q^{28} - 6 q^{29} + 8 q^{32} + 20 q^{33} + 16 q^{36} + 40 q^{37} - 6 q^{47} + 30 q^{49} + 20 q^{51} + 10 q^{56}+ \cdots + 30 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 + 10 * q^7 + 8 * q^8 + 16 * q^9 + 10 * q^14 + 8 * q^16 + 16 * q^18 + 10 * q^28 - 6 * q^29 + 8 * q^32 + 20 * q^33 + 16 * q^36 + 40 * q^37 - 6 * q^47 + 30 * q^49 + 20 * q^51 + 10 * q^56 + 24 * q^57 - 6 * q^58 + 10 * q^61 + 50 * q^63 + 8 * q^64 + 20 * q^66 + 32 * q^67 + 4 * q^69 + 16 * q^72 + 26 * q^73 + 40 * q^74 - 4 * q^79 - 16 * q^81 + 48 * q^83 + 36 * q^93 - 6 * q^94 + 8 * q^97 + 30 * q^98

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