LMFDB - Embedded newform 8281.2.a.cx.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 637)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	8281.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.04004 q^{2} +0.769363 q^{3} -0.918323 q^{4} +1.67669 q^{5} -0.800166 q^{6} +3.03516 q^{8} -2.40808 q^{9} +O(q^{10})$	\(q-1.04004 q^{2} +0.769363 q^{3} -0.918323 q^{4} +1.67669 q^{5} -0.800166 q^{6} +3.03516 q^{8} -2.40808 q^{9} -1.74382 q^{10} -0.537414 q^{11} -0.706523 q^{12} +1.28998 q^{15} -1.32004 q^{16} +5.62885 q^{17} +2.50449 q^{18} -2.01436 q^{19} -1.53974 q^{20} +0.558930 q^{22} -6.67561 q^{23} +2.33514 q^{24} -2.18871 q^{25} -4.16078 q^{27} -4.87421 q^{29} -1.34163 q^{30} +3.78899 q^{31} -4.69744 q^{32} -0.413466 q^{33} -5.85422 q^{34} +2.21140 q^{36} -1.38192 q^{37} +2.09501 q^{38} +5.08903 q^{40} -1.24156 q^{41} -3.26753 q^{43} +0.493519 q^{44} -4.03760 q^{45} +6.94288 q^{46} +3.79963 q^{47} -1.01559 q^{48} +2.27634 q^{50} +4.33063 q^{51} +13.4665 q^{53} +4.32736 q^{54} -0.901076 q^{55} -1.54978 q^{57} +5.06936 q^{58} +8.07619 q^{59} -1.18462 q^{60} +3.77824 q^{61} -3.94069 q^{62} +7.52559 q^{64} +0.430020 q^{66} -9.59752 q^{67} -5.16910 q^{68} -5.13597 q^{69} +1.52081 q^{71} -7.30892 q^{72} +15.6451 q^{73} +1.43725 q^{74} -1.68391 q^{75} +1.84984 q^{76} +8.26567 q^{79} -2.21329 q^{80} +4.02310 q^{81} +1.29127 q^{82} -9.42392 q^{83} +9.43784 q^{85} +3.39835 q^{86} -3.75004 q^{87} -1.63114 q^{88} +2.13967 q^{89} +4.19926 q^{90} +6.13037 q^{92} +2.91511 q^{93} -3.95175 q^{94} -3.37746 q^{95} -3.61403 q^{96} +6.44451 q^{97} +1.29414 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 40 q^{4} + 56 q^{9}+O(q^{10}) $ 32 * q + 40 * q^4 + 56 * q^9	$ 32 q + 40 q^{4} + 56 q^{9} + 56 q^{16} - 16 q^{22} + 48 q^{23} + 40 q^{25} + 48 q^{29} + 48 q^{30} + 184 q^{36} + 24 q^{43} + 72 q^{53} - 32 q^{64} - 48 q^{74} + 96 q^{79} + 128 q^{81} + 112 q^{88} + 168 q^{92} + 168 q^{95}+O(q^{100}) $ 32 * q + 40 * q^4 + 56 * q^9 + 56 * q^16 - 16 * q^22 + 48 * q^23 + 40 * q^25 + 48 * q^29 + 48 * q^30 + 184 * q^36 + 24 * q^43 + 72 * q^53 - 32 * q^64 - 48 * q^74 + 96 * q^79 + 128 * q^81 + 112 * q^88 + 168 * q^92 + 168 * q^95

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 $

$2$	−1.04004	−0.735417	−0.367709	−	0.929941i	$-0.619858\pi$
$2$	−1.04004	−0.735417	−0.367709	+	0.929941i	$0.619858\pi$
$3$	0.769363	0.444192	0.222096	−	0.975025i	$-0.428710\pi$
$3$	0.769363	0.444192	0.222096	+	0.975025i	$0.428710\pi$
$4$	−0.918323	−0.459161
$5$	1.67669	0.749838	0.374919	−	0.927057i	$-0.377670\pi$
$5$	1.67669	0.749838	0.374919	+	0.927057i	$0.377670\pi$
$6$	−0.800166	−0.326666
$7$	0	0
$7$	0	0
$8$	3.03516	1.07309
$9$	−2.40808	−0.802694
$10$	−1.74382	−0.551444
$11$	−0.537414	−0.162036	−0.0810182	−	0.996713i	$-0.525817\pi$
$11$	−0.537414	−0.162036	−0.0810182	+	0.996713i	$0.525817\pi$
$12$	−0.706523	−0.203956
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.28998	0.333072
$16$	−1.32004	−0.330009
$17$	5.62885	1.36520	0.682599	−	0.730793i	$-0.260850\pi$
$17$	5.62885	1.36520	0.682599	+	0.730793i	$0.260850\pi$
$18$	2.50449	0.590315
$19$	−2.01436	−0.462127	−0.231063	−	0.972939i	$-0.574220\pi$
$19$	−2.01436	−0.462127	−0.231063	+	0.972939i	$0.574220\pi$
$20$	−1.53974	−0.344297
$21$	0	0
$22$	0.558930	0.119164
$23$	−6.67561	−1.39196	−0.695981	−	0.718061i	$-0.745030\pi$
$23$	−6.67561	−1.39196	−0.695981	+	0.718061i	$0.745030\pi$
$24$	2.33514	0.476659
$25$	−2.18871	−0.437742
$26$	0	0
$27$	−4.16078	−0.800742
$28$	0	0
$29$	−4.87421	−0.905119	−0.452559	−	0.891734i	$-0.649489\pi$
$29$	−4.87421	−0.905119	−0.452559	+	0.891734i	$0.649489\pi$
$30$	−1.34163	−0.244947
$31$	3.78899	0.680523	0.340261	−	0.940331i	$-0.389484\pi$
$31$	3.78899	0.680523	0.340261	+	0.940331i	$0.389484\pi$
$32$	−4.69744	−0.830398
$33$	−0.413466	−0.0719752
$34$	−5.85422	−1.00399
$35$	0	0
$36$	2.21140	0.368566
$37$	−1.38192	−0.227187	−0.113593	−	0.993527i	$-0.536236\pi$
$37$	−1.38192	−0.227187	−0.113593	+	0.993527i	$0.536236\pi$
$38$	2.09501	0.339856
$39$	0	0
$40$	5.08903	0.804646
$41$	−1.24156	−0.193900	−0.0969498	−	0.995289i	$-0.530909\pi$
$41$	−1.24156	−0.193900	−0.0969498	+	0.995289i	$0.530909\pi$
$42$	0	0
$43$	−3.26753	−0.498294	−0.249147	−	0.968466i	$-0.580150\pi$
$43$	−3.26753	−0.498294	−0.249147	+	0.968466i	$0.580150\pi$
$44$	0.493519	0.0744008
$45$	−4.03760	−0.601891
$46$	6.94288	1.02367
$47$	3.79963	0.554232	0.277116	−	0.960836i	$-0.410621\pi$
$47$	3.79963	0.554232	0.277116	+	0.960836i	$0.410621\pi$
$48$	−1.01559	−0.146587
$49$	0	0
$50$	2.27634	0.321923
$51$	4.33063	0.606409
$52$	0	0
$53$	13.4665	1.84977	0.924886	−	0.380246i	$-0.124161\pi$
$53$	13.4665	1.84977	0.924886	+	0.380246i	$0.124161\pi$
$54$	4.32736	0.588879
$55$	−0.901076	−0.121501
$56$	0	0
$57$	−1.54978	−0.205273
$58$	5.06936	0.665640
$59$	8.07619	1.05143	0.525715	−	0.850661i	$-0.323798\pi$
$59$	8.07619	1.05143	0.525715	+	0.850661i	$0.323798\pi$
$60$	−1.18462	−0.152934
$61$	3.77824	0.483754	0.241877	−	0.970307i	$-0.422237\pi$
$61$	3.77824	0.483754	0.241877	+	0.970307i	$0.422237\pi$
$62$	−3.94069	−0.500468
$63$	0	0
$64$	7.52559	0.940698
$65$	0	0
$66$	0.430020	0.0529318
$67$	−9.59752	−1.17252	−0.586262	−	0.810121i	$-0.699401\pi$
$67$	−9.59752	−1.17252	−0.586262	+	0.810121i	$0.699401\pi$
$68$	−5.16910	−0.626846
$69$	−5.13597	−0.618298
$70$	0	0
$71$	1.52081	0.180487	0.0902433	−	0.995920i	$-0.471236\pi$
$71$	1.52081	0.180487	0.0902433	+	0.995920i	$0.471236\pi$
$72$	−7.30892	−0.861365
$73$	15.6451	1.83112	0.915560	−	0.402181i	$-0.131748\pi$
$73$	15.6451	1.83112	0.915560	+	0.402181i	$0.131748\pi$
$74$	1.43725	0.167077
$75$	−1.68391	−0.194442
$76$	1.84984	0.212191
$77$	0	0
$78$	0	0
$79$	8.26567	0.929961	0.464980	−	0.885321i	$-0.346061\pi$
$79$	8.26567	0.929961	0.464980	+	0.885321i	$0.346061\pi$
$80$	−2.21329	−0.247454
$81$	4.02310	0.447011
$82$	1.29127	0.142597
$83$	−9.42392	−1.03441	−0.517205	−	0.855862i	$-0.673027\pi$
$83$	−9.42392	−1.03441	−0.517205	+	0.855862i	$0.673027\pi$
$84$	0	0
$85$	9.43784	1.02368
$86$	3.39835	0.366454
$87$	−3.75004	−0.402046
$88$	−1.63114	−0.173880
$89$	2.13967	0.226804	0.113402	−	0.993549i	$-0.463825\pi$
$89$	2.13967	0.226804	0.113402	+	0.993549i	$0.463825\pi$
$90$	4.19926	0.442641
$91$	0	0
$92$	6.13037	0.639135
$93$	2.91511	0.302283
$94$	−3.95175	−0.407592
$95$	−3.37746	−0.346520
$96$	−3.61403	−0.368856
$97$	6.44451	0.654341	0.327170	−	0.944965i	$-0.393905\pi$
$97$	6.44451	0.654341	0.327170	+	0.944965i	$0.393905\pi$
$98$	0	0
$99$	1.29414	0.130066
$100$	2.00994	0.200994
$101$	7.38945	0.735278	0.367639	−	0.929969i	$-0.380166\pi$
$101$	7.38945	0.735278	0.367639	+	0.929969i	$0.380166\pi$
$102$	−4.50402	−0.445964
$103$	17.5253	1.72682	0.863411	−	0.504502i	$-0.168324\pi$
$103$	17.5253	1.72682	0.863411	+	0.504502i	$0.168324\pi$
$104$	0	0
$105$	0	0
$106$	−14.0057	−1.36035
$107$	5.02309	0.485600	0.242800	−	0.970076i	$-0.421934\pi$
$107$	5.02309	0.485600	0.242800	+	0.970076i	$0.421934\pi$
$108$	3.82094	0.367670
$109$	−17.8660	−1.71125	−0.855626	−	0.517595i	$-0.826827\pi$
$109$	−17.8660	−1.71125	−0.855626	+	0.517595i	$0.826827\pi$
$110$	0.937153	0.0893540
$111$	−1.06320	−0.100914
$112$	0	0
$113$	−7.57506	−0.712602	−0.356301	−	0.934371i	$-0.615962\pi$
$113$	−7.57506	−0.712602	−0.356301	+	0.934371i	$0.615962\pi$
$114$	1.61182	0.150961
$115$	−11.1929	−1.04375
$116$	4.47610	0.415596
$117$	0	0
$118$	−8.39954	−0.773240
$119$	0	0
$120$	3.91531	0.357417
$121$	−10.7112	−0.973744
$122$	−3.92951	−0.355761
$123$	−0.955212	−0.0861286
$124$	−3.47952	−0.312470
$125$	−12.0532	−1.07807
$126$	0	0
$127$	−10.6721	−0.946995	−0.473497	−	0.880795i	$-0.657009\pi$
$127$	−10.6721	−0.946995	−0.473497	+	0.880795i	$0.657009\pi$
$128$	1.56799	0.138592
$129$	−2.51392	−0.221338
$130$	0	0
$131$	9.01437	0.787589	0.393794	−	0.919199i	$-0.371162\pi$
$131$	9.01437	0.787589	0.393794	+	0.919199i	$0.371162\pi$
$132$	0.379695	0.0330482
$133$	0	0
$134$	9.98178	0.862294
$135$	−6.97633	−0.600427
$136$	17.0845	1.46498
$137$	−4.12950	−0.352807	−0.176404	−	0.984318i	$-0.556446\pi$
$137$	−4.12950	−0.352807	−0.176404	+	0.984318i	$0.556446\pi$
$138$	5.34160	0.454707
$139$	19.7115	1.67191	0.835955	−	0.548798i	$-0.184915\pi$
$139$	19.7115	1.67191	0.835955	+	0.548798i	$0.184915\pi$
$140$	0	0
$141$	2.92329	0.246185
$142$	−1.58170	−0.132733
$143$	0	0
$144$	3.17876	0.264897
$145$	−8.17255	−0.678693
$146$	−16.2715	−1.34664
$147$	0	0
$148$	1.26905	0.104315
$149$	−14.7880	−1.21148	−0.605740	−	0.795663i	$-0.707123\pi$
$149$	−14.7880	−1.21148	−0.605740	+	0.795663i	$0.707123\pi$
$150$	1.75133	0.142996
$151$	16.1538	1.31458	0.657289	−	0.753639i	$-0.271703\pi$
$151$	16.1538	1.31458	0.657289	+	0.753639i	$0.271703\pi$
$152$	−6.11392	−0.495905
$153$	−13.5547	−1.09584
$154$	0	0
$155$	6.35296	0.510282
$156$	0	0
$157$	−2.91745	−0.232838	−0.116419	−	0.993200i	$-0.537141\pi$
$157$	−2.91745	−0.232838	−0.116419	+	0.993200i	$0.537141\pi$
$158$	−8.59661	−0.683909
$159$	10.3607	0.821653
$160$	−7.87615	−0.622664
$161$	0	0
$162$	−4.18417	−0.328739
$163$	−16.8393	−1.31896	−0.659479	−	0.751723i	$-0.729223\pi$
$163$	−16.8393	−1.31896	−0.659479	+	0.751723i	$0.729223\pi$
$164$	1.14016	0.0890312
$165$	−0.693254	−0.0539698
$166$	9.80122	0.760722
$167$	24.8739	1.92480	0.962399	−	0.271639i	$-0.0875657\pi$
$167$	24.8739	1.92480	0.962399	+	0.271639i	$0.0875657\pi$
$168$	0	0
$169$	0	0
$170$	−9.81571	−0.752830
$171$	4.85075	0.370946
$172$	3.00065	0.228797
$173$	−0.155531	−0.0118248	−0.00591239	−	0.999983i	$-0.501882\pi$
$173$	−0.155531	−0.0118248	−0.00591239	+	0.999983i	$0.501882\pi$
$174$	3.90018	0.295672
$175$	0	0
$176$	0.709407	0.0534735
$177$	6.21352	0.467037
$178$	−2.22533	−0.166796
$179$	20.4246	1.52660	0.763302	−	0.646042i	$-0.223577\pi$
$179$	20.4246	1.52660	0.763302	+	0.646042i	$0.223577\pi$
$180$	3.70782	0.276365
$181$	−5.78777	−0.430202	−0.215101	−	0.976592i	$-0.569008\pi$
$181$	−5.78777	−0.430202	−0.215101	+	0.976592i	$0.569008\pi$
$182$	0	0
$183$	2.90684	0.214880
$184$	−20.2616	−1.49370
$185$	−2.31706	−0.170353
$186$	−3.03182	−0.222304
$187$	−3.02502	−0.221212
$188$	−3.48928	−0.254482
$189$	0	0
$190$	3.51269	0.254837
$191$	−8.45850	−0.612036	−0.306018	−	0.952026i	$-0.598997\pi$
$191$	−8.45850	−0.612036	−0.306018	+	0.952026i	$0.598997\pi$
$192$	5.78991	0.417850
$193$	−1.34663	−0.0969326	−0.0484663	−	0.998825i	$-0.515433\pi$
$193$	−1.34663	−0.0969326	−0.0484663	+	0.998825i	$0.515433\pi$
$194$	−6.70253	−0.481214
$195$	0	0
$196$	0	0
$197$	−6.39568	−0.455673	−0.227837	−	0.973699i	$-0.573165\pi$
$197$	−6.39568	−0.455673	−0.227837	+	0.973699i	$0.573165\pi$
$198$	−1.34595	−0.0956525
$199$	17.5430	1.24359	0.621794	−	0.783180i	$-0.286404\pi$
$199$	17.5430	1.24359	0.621794	+	0.783180i	$0.286404\pi$
$200$	−6.64310	−0.469738
$201$	−7.38398	−0.520826
$202$	−7.68531	−0.540736
$203$	0	0
$204$	−3.97692	−0.278440
$205$	−2.08172	−0.145393
$206$	−18.2270	−1.26993
$207$	16.0754	1.11732
$208$	0	0
$209$	1.08255	0.0748813
$210$	0	0
$211$	19.7953	1.36277	0.681383	−	0.731927i	$-0.261379\pi$
$211$	19.7953	1.36277	0.681383	+	0.731927i	$0.261379\pi$
$212$	−12.3666	−0.849343
$213$	1.17005	0.0801706
$214$	−5.22420	−0.357119
$215$	−5.47863	−0.373640
$216$	−12.6286	−0.859270
$217$	0	0
$218$	18.5813	1.25848
$219$	12.0368	0.813369
$220$	0.827479	0.0557886
$221$	0	0
$222$	1.10577	0.0742142
$223$	26.8537	1.79826	0.899128	−	0.437685i	$-0.144202\pi$
$223$	26.8537	1.79826	0.899128	+	0.437685i	$0.144202\pi$
$224$	0	0
$225$	5.27059	0.351373
$226$	7.87835	0.524060
$227$	1.64115	0.108927	0.0544634	−	0.998516i	$-0.482655\pi$
$227$	1.64115	0.108927	0.0544634	+	0.998516i	$0.482655\pi$
$228$	1.42319	0.0942534
$229$	−6.98091	−0.461311	−0.230656	−	0.973035i	$-0.574087\pi$
$229$	−6.98091	−0.461311	−0.230656	+	0.973035i	$0.574087\pi$
$230$	11.6411	0.767589
$231$	0	0
$232$	−14.7940	−0.971276
$233$	14.3619	0.940882	0.470441	−	0.882431i	$-0.344095\pi$
$233$	14.3619	0.940882	0.470441	+	0.882431i	$0.344095\pi$
$234$	0	0
$235$	6.37079	0.415585
$236$	−7.41655	−0.482776
$237$	6.35930	0.413081
$238$	0	0
$239$	−19.9962	−1.29345	−0.646724	−	0.762724i	$-0.723861\pi$
$239$	−19.9962	−1.29345	−0.646724	+	0.762724i	$0.723861\pi$
$240$	−1.70283	−0.109917
$241$	27.7018	1.78443	0.892214	−	0.451613i	$-0.149151\pi$
$241$	27.7018	1.78443	0.892214	+	0.451613i	$0.149151\pi$
$242$	11.1400	0.716108
$243$	15.5775	0.999300
$244$	−3.46965	−0.222121
$245$	0	0
$246$	0.993456	0.0633405
$247$	0	0
$248$	11.5002	0.730264
$249$	−7.25041	−0.459476
$250$	12.5358	0.792835
$251$	−9.59564	−0.605671	−0.302836	−	0.953043i	$-0.597933\pi$
$251$	−9.59564	−0.605671	−0.302836	+	0.953043i	$0.597933\pi$
$252$	0	0
$253$	3.58757	0.225548
$254$	11.0994	0.696436
$255$	7.26112	0.454709
$256$	−16.6819	−1.04262
$257$	4.91265	0.306442	0.153221	−	0.988192i	$-0.451035\pi$
$257$	4.91265	0.306442	0.153221	+	0.988192i	$0.451035\pi$
$258$	2.61457	0.162776
$259$	0	0
$260$	0	0
$261$	11.7375	0.726533
$262$	−9.37528	−0.579206
$263$	18.5372	1.14305	0.571526	−	0.820584i	$-0.306351\pi$
$263$	18.5372	1.14305	0.571526	+	0.820584i	$0.306351\pi$
$264$	−1.25494	−0.0772361
$265$	22.5792	1.38703
$266$	0	0
$267$	1.64618	0.100745
$268$	8.81362	0.538378
$269$	−13.7840	−0.840427	−0.420213	−	0.907425i	$-0.638045\pi$
$269$	−13.7840	−0.840427	−0.420213	+	0.907425i	$0.638045\pi$
$270$	7.25564	0.441564
$271$	−19.9963	−1.21469	−0.607343	−	0.794439i	$-0.707765\pi$
$271$	−19.9963	−1.21469	−0.607343	+	0.794439i	$0.707765\pi$
$272$	−7.43030	−0.450528
$273$	0	0
$274$	4.29484	0.259461
$275$	1.17624	0.0709302
$276$	4.71647	0.283898
$277$	25.8817	1.55508	0.777540	−	0.628833i	$-0.216467\pi$
$277$	25.8817	1.55508	0.777540	+	0.628833i	$0.216467\pi$
$278$	−20.5007	−1.22955
$279$	−9.12419	−0.546251
$280$	0	0
$281$	20.2430	1.20760	0.603798	−	0.797138i	$-0.293653\pi$
$281$	20.2430	1.20760	0.603798	+	0.797138i	$0.293653\pi$
$282$	−3.04033	−0.181049
$283$	22.3954	1.33127	0.665635	−	0.746277i	$-0.268161\pi$
$283$	22.3954	1.33127	0.665635	+	0.746277i	$0.268161\pi$
$284$	−1.39659	−0.0828724
$285$	−2.59849	−0.153921
$286$	0	0
$287$	0	0
$288$	11.3118	0.666555
$289$	14.6840	0.863764
$290$	8.49975	0.499123
$291$	4.95817	0.290653
$292$	−14.3673	−0.840780
$293$	−22.9223	−1.33914	−0.669568	−	0.742751i	$-0.733521\pi$
$293$	−22.9223	−1.33914	−0.669568	+	0.742751i	$0.733521\pi$
$294$	0	0
$295$	13.5413	0.788403
$296$	−4.19436	−0.243792
$297$	2.23606	0.129749
$298$	15.3801	0.890943
$299$	0	0
$300$	1.54638	0.0892800
$301$	0	0
$302$	−16.8006	−0.966763
$303$	5.68517	0.326605
$304$	2.65904	0.152506
$305$	6.33494	0.362738
$306$	14.0974	0.805896
$307$	−28.9163	−1.65034	−0.825170	−	0.564884i	$-0.808921\pi$
$307$	−28.9163	−1.65034	−0.825170	+	0.564884i	$0.808921\pi$
$308$	0	0
$309$	13.4833	0.767040
$310$	−6.60731	−0.375270
$311$	−5.23639	−0.296928	−0.148464	−	0.988918i	$-0.547433\pi$
$311$	−5.23639	−0.296928	−0.148464	+	0.988918i	$0.547433\pi$
$312$	0	0
$313$	15.7163	0.888338	0.444169	−	0.895943i	$-0.353499\pi$
$313$	15.7163	0.888338	0.444169	+	0.895943i	$0.353499\pi$
$314$	3.03425	0.171233
$315$	0	0
$316$	−7.59056	−0.427002
$317$	−25.3794	−1.42545	−0.712725	−	0.701444i	$-0.752539\pi$
$317$	−25.3794	−1.42545	−0.712725	+	0.701444i	$0.752539\pi$
$318$	−10.7755	−0.604258
$319$	2.61947	0.146662
$320$	12.6181	0.705372
$321$	3.86458	0.215700
$322$	0	0
$323$	−11.3386	−0.630894
$324$	−3.69450	−0.205250
$325$	0	0
$326$	17.5135	0.969984
$327$	−13.7454	−0.760124
$328$	−3.76835	−0.208072
$329$	0	0
$330$	0.721010	0.0396903
$331$	14.2896	0.785426	0.392713	−	0.919661i	$-0.371537\pi$
$331$	14.2896	0.785426	0.392713	+	0.919661i	$0.371537\pi$
$332$	8.65420	0.474961
$333$	3.32778	0.182361
$334$	−25.8698	−1.41553
$335$	−16.0921	−0.879204
$336$	0	0
$337$	−7.63229	−0.415757	−0.207879	−	0.978155i	$-0.566656\pi$
$337$	−7.63229	−0.415757	−0.207879	+	0.978155i	$0.566656\pi$
$338$	0	0
$339$	−5.82797	−0.316532
$340$	−8.66698	−0.470033
$341$	−2.03626	−0.110269
$342$	−5.04496	−0.272800
$343$	0	0
$344$	−9.91749	−0.534715
$345$	−8.61142	−0.463623
$346$	0.161758	0.00869615
$347$	−12.2689	−0.658631	−0.329315	−	0.944220i	$-0.606818\pi$
$347$	−12.2689	−0.658631	−0.329315	+	0.944220i	$0.606818\pi$
$348$	3.44375	0.184604
$349$	16.8138	0.900020	0.450010	−	0.893023i	$-0.351420\pi$
$349$	16.8138	0.900020	0.450010	+	0.893023i	$0.351420\pi$
$350$	0	0
$351$	0	0
$352$	2.52447	0.134555
$353$	−11.1223	−0.591982	−0.295991	−	0.955191i	$-0.595650\pi$
$353$	−11.1223	−0.591982	−0.295991	+	0.955191i	$0.595650\pi$
$354$	−6.46229	−0.343467
$355$	2.54992	0.135336
$356$	−1.96490	−0.104140
$357$	0	0
$358$	−21.2423	−1.12269
$359$	−34.3599	−1.81345	−0.906723	−	0.421727i	$-0.861424\pi$
$359$	−34.3599	−1.81345	−0.906723	+	0.421727i	$0.861424\pi$
$360$	−12.2548	−0.645884
$361$	−14.9423	−0.786439
$362$	6.01950	0.316378
$363$	−8.24079	−0.432529
$364$	0	0
$365$	26.2320	1.37304
$366$	−3.02322	−0.158026
$367$	22.5496	1.17708	0.588539	−	0.808469i	$-0.299703\pi$
$367$	22.5496	1.17708	0.588539	+	0.808469i	$0.299703\pi$
$368$	8.81206	0.459360
$369$	2.98978	0.155642
$370$	2.40982	0.125281
$371$	0	0
$372$	−2.67701	−0.138796
$373$	12.1437	0.628778	0.314389	−	0.949294i	$-0.398200\pi$
$373$	12.1437	0.628778	0.314389	+	0.949294i	$0.398200\pi$
$374$	3.14614	0.162683
$375$	−9.27331	−0.478872
$376$	11.5325	0.594743
$377$	0	0
$378$	0	0
$379$	−15.7907	−0.811116	−0.405558	−	0.914069i	$-0.632923\pi$
$379$	−15.7907	−0.811116	−0.405558	+	0.914069i	$0.632923\pi$
$380$	3.10160	0.159109
$381$	−8.21071	−0.420647
$382$	8.79716	0.450102
$383$	26.0571	1.33146	0.665728	−	0.746194i	$-0.268121\pi$
$383$	26.0571	1.33146	0.665728	+	0.746194i	$0.268121\pi$
$384$	1.20635	0.0615614
$385$	0	0
$386$	1.40055	0.0712859
$387$	7.86848	0.399977
$388$	−5.91814	−0.300448
$389$	12.9605	0.657122	0.328561	−	0.944483i	$-0.393436\pi$
$389$	12.9605	0.657122	0.328561	+	0.944483i	$0.393436\pi$
$390$	0	0
$391$	−37.5760	−1.90030
$392$	0	0
$393$	6.93532	0.349840
$394$	6.65174	0.335110
$395$	13.8590	0.697321
$396$	−1.18843	−0.0597211
$397$	9.63056	0.483344	0.241672	−	0.970358i	$-0.422304\pi$
$397$	9.63056	0.483344	0.241672	+	0.970358i	$0.422304\pi$
$398$	−18.2454	−0.914557
$399$	0	0
$400$	2.88918	0.144459
$401$	−11.3719	−0.567887	−0.283943	−	0.958841i	$-0.591643\pi$
$401$	−11.3719	−0.567887	−0.283943	+	0.958841i	$0.591643\pi$
$402$	7.67961	0.383024
$403$	0	0
$404$	−6.78590	−0.337611
$405$	6.74549	0.335186
$406$	0	0
$407$	0.742664	0.0368125
$408$	13.1442	0.650734
$409$	−21.6918	−1.07259	−0.536294	−	0.844031i	$-0.680176\pi$
$409$	−21.6918	−1.07259	−0.536294	+	0.844031i	$0.680176\pi$
$410$	2.16506	0.106925
$411$	−3.17709	−0.156714
$412$	−16.0939	−0.792890
$413$	0	0
$414$	−16.7190	−0.821695
$415$	−15.8010	−0.775640
$416$	0	0
$417$	15.1653	0.742648
$418$	−1.12589	−0.0550690
$419$	−5.91593	−0.289012	−0.144506	−	0.989504i	$-0.546159\pi$
$419$	−5.91593	−0.289012	−0.144506	+	0.989504i	$0.546159\pi$
$420$	0	0
$421$	−1.39103	−0.0677946	−0.0338973	−	0.999425i	$-0.510792\pi$
$421$	−1.39103	−0.0677946	−0.0338973	+	0.999425i	$0.510792\pi$
$422$	−20.5879	−1.00220
$423$	−9.14981	−0.444879
$424$	40.8731	1.98498
$425$	−12.3199	−0.597605
$426$	−1.21690	−0.0589589
$427$	0	0
$428$	−4.61282	−0.222969
$429$	0	0
$430$	5.69798	0.274781
$431$	5.03674	0.242611	0.121306	−	0.992615i	$-0.461292\pi$
$431$	5.03674	0.242611	0.121306	+	0.992615i	$0.461292\pi$
$432$	5.49238	0.264252
$433$	−14.7409	−0.708403	−0.354201	−	0.935169i	$-0.615247\pi$
$433$	−14.7409	−0.708403	−0.354201	+	0.935169i	$0.615247\pi$
$434$	0	0
$435$	−6.28765	−0.301470
$436$	16.4067	0.785740
$437$	13.4471	0.643262
$438$	−12.5187	−0.598165
$439$	−7.65827	−0.365509	−0.182755	−	0.983159i	$-0.558501\pi$
$439$	−7.65827	−0.365509	−0.182755	+	0.983159i	$0.558501\pi$
$440$	−2.73491	−0.130382
$441$	0	0
$442$	0	0
$443$	30.0570	1.42805	0.714025	−	0.700121i	$-0.246870\pi$
$443$	30.0570	1.42805	0.714025	+	0.700121i	$0.246870\pi$
$444$	0.976361	0.0463360
$445$	3.58756	0.170066
$446$	−27.9288	−1.32247
$447$	−11.3773	−0.538129
$448$	0	0
$449$	33.1263	1.56333	0.781664	−	0.623700i	$-0.214371\pi$
$449$	33.1263	1.56333	0.781664	+	0.623700i	$0.214371\pi$
$450$	−5.48161	−0.258406
$451$	0.667233	0.0314188
$452$	6.95635	0.327199
$453$	12.4281	0.583925
$454$	−1.70686	−0.0801067
$455$	0	0
$456$	−4.70382	−0.220277
$457$	22.0538	1.03163	0.515817	−	0.856699i	$-0.327488\pi$
$457$	22.0538	1.03163	0.515817	+	0.856699i	$0.327488\pi$
$458$	7.26040	0.339256
$459$	−23.4204	−1.09317
$460$	10.2787	0.479248
$461$	22.0189	1.02552	0.512761	−	0.858531i	$-0.328623\pi$
$461$	22.0189	1.02552	0.512761	+	0.858531i	$0.328623\pi$
$462$	0	0
$463$	35.0344	1.62819	0.814093	−	0.580734i	$-0.197234\pi$
$463$	35.0344	1.62819	0.814093	+	0.580734i	$0.197234\pi$
$464$	6.43415	0.298698
$465$	4.88773	0.226663
$466$	−14.9370	−0.691941
$467$	−19.0158	−0.879948	−0.439974	−	0.898010i	$-0.645012\pi$
$467$	−19.0158	−0.879948	−0.439974	+	0.898010i	$0.645012\pi$
$468$	0	0
$469$	0	0
$470$	−6.62586	−0.305628
$471$	−2.24457	−0.103425
$472$	24.5126	1.12828
$473$	1.75602	0.0807417
$474$	−6.61391	−0.303787
$475$	4.40886	0.202292
$476$	0	0
$477$	−32.4285	−1.48480
$478$	20.7968	0.951224
$479$	20.3100	0.927987	0.463994	−	0.885839i	$-0.346416\pi$
$479$	20.3100	0.927987	0.463994	+	0.885839i	$0.346416\pi$
$480$	−6.05962	−0.276582
$481$	0	0
$482$	−28.8109	−1.31230
$483$	0	0
$484$	9.83633	0.447106
$485$	10.8054	0.490650
$486$	−16.2012	−0.734903
$487$	−6.43469	−0.291584	−0.145792	−	0.989315i	$-0.546573\pi$
$487$	−6.43469	−0.291584	−0.145792	+	0.989315i	$0.546573\pi$
$488$	11.4676	0.519113
$489$	−12.9556	−0.585870
$490$	0	0
$491$	0.960427	0.0433434	0.0216717	−	0.999765i	$-0.493101\pi$
$491$	0.960427	0.0433434	0.0216717	+	0.999765i	$0.493101\pi$
$492$	0.877193	0.0395469
$493$	−27.4362	−1.23567
$494$	0	0
$495$	2.16986	0.0975282
$496$	−5.00161	−0.224579
$497$	0	0
$498$	7.54069	0.337907
$499$	9.78848	0.438192	0.219096	−	0.975703i	$-0.429689\pi$
$499$	9.78848	0.438192	0.219096	+	0.975703i	$0.429689\pi$
$500$	11.0688	0.495010
$501$	19.1370	0.854980
$502$	9.97982	0.445421
$503$	12.1169	0.540266	0.270133	−	0.962823i	$-0.412932\pi$
$503$	12.1169	0.540266	0.270133	+	0.962823i	$0.412932\pi$
$504$	0	0
$505$	12.3898	0.551340
$506$	−3.73120	−0.165872
$507$	0	0
$508$	9.80042	0.434823
$509$	−30.9382	−1.37131	−0.685656	−	0.727926i	$-0.740485\pi$
$509$	−30.9382	−1.37131	−0.685656	+	0.727926i	$0.740485\pi$
$510$	−7.55184	−0.334401
$511$	0	0
$512$	14.2139	0.628170
$513$	8.38131	0.370044
$514$	−5.10933	−0.225363
$515$	29.3845	1.29484
$516$	2.30859	0.101630
$517$	−2.04197	−0.0898058
$518$	0	0
$519$	−0.119660	−0.00525247
$520$	0	0
$521$	35.0676	1.53634	0.768169	−	0.640247i	$-0.221168\pi$
$521$	35.0676	1.53634	0.768169	+	0.640247i	$0.221168\pi$
$522$	−12.2074	−0.534305
$523$	−34.4475	−1.50628	−0.753142	−	0.657858i	$-0.771463\pi$
$523$	−34.4475	−1.50628	−0.753142	+	0.657858i	$0.771463\pi$
$524$	−8.27810	−0.361630
$525$	0	0
$526$	−19.2794	−0.840621
$527$	21.3277	0.929048
$528$	0.545791	0.0237525
$529$	21.5638	0.937556
$530$	−23.4832	−1.02005
$531$	−19.4481	−0.843977
$532$	0	0
$533$	0	0
$534$	−1.71209	−0.0740893
$535$	8.42216	0.364122
$536$	−29.1301	−1.25823
$537$	15.7139	0.678105
$538$	14.3359	0.618064
$539$	0	0
$540$	6.40652	0.275693
$541$	9.77797	0.420388	0.210194	−	0.977660i	$-0.432590\pi$
$541$	9.77797	0.420388	0.210194	+	0.977660i	$0.432590\pi$
$542$	20.7969	0.893302
$543$	−4.45289	−0.191092
$544$	−26.4412	−1.13366
$545$	−29.9557	−1.28316
$546$	0	0
$547$	19.7876	0.846057	0.423029	−	0.906116i	$-0.360967\pi$
$547$	19.7876	0.846057	0.423029	+	0.906116i	$0.360967\pi$
$548$	3.79222	0.161995
$549$	−9.09831	−0.388307
$550$	−1.22334	−0.0521633
$551$	9.81844	0.418280
$552$	−15.5885	−0.663491
$553$	0	0
$554$	−26.9179	−1.14363
$555$	−1.78266	−0.0756695
$556$	−18.1015	−0.767676
$557$	1.26714	0.0536905	0.0268452	−	0.999640i	$-0.491454\pi$
$557$	1.26714	0.0536905	0.0268452	+	0.999640i	$0.491454\pi$
$558$	9.48950	0.401723
$559$	0	0
$560$	0	0
$561$	−2.32734	−0.0982604
$562$	−21.0535	−0.888087
$563$	−39.8317	−1.67870	−0.839352	−	0.543588i	$-0.817066\pi$
$563$	−39.8317	−1.67870	−0.839352	+	0.543588i	$0.817066\pi$
$564$	−2.68452	−0.113039
$565$	−12.7010	−0.534336
$566$	−23.2921	−0.979039
$567$	0	0
$568$	4.61590	0.193679
$569$	5.46902	0.229273	0.114637	−	0.993407i	$-0.463430\pi$
$569$	5.46902	0.229273	0.114637	+	0.993407i	$0.463430\pi$
$570$	2.70253	0.113197
$571$	−4.67594	−0.195682	−0.0978411	−	0.995202i	$-0.531194\pi$
$571$	−4.67594	−0.195682	−0.0978411	+	0.995202i	$0.531194\pi$
$572$	0	0
$573$	−6.50766	−0.271861
$574$	0	0
$575$	14.6110	0.609320
$576$	−18.1222	−0.755093
$577$	−3.60435	−0.150051	−0.0750255	−	0.997182i	$-0.523904\pi$
$577$	−3.60435	−0.150051	−0.0750255	+	0.997182i	$0.523904\pi$
$578$	−15.2719	−0.635227
$579$	−1.03605	−0.0430567
$580$	7.50503	0.311630
$581$	0	0
$582$	−5.15668	−0.213751
$583$	−7.23710	−0.299730
$584$	47.4854	1.96496
$585$	0	0
$586$	23.8400	0.984823
$587$	23.4720	0.968795	0.484398	−	0.874848i	$-0.339039\pi$
$587$	23.4720	0.968795	0.484398	+	0.874848i	$0.339039\pi$
$588$	0	0
$589$	−7.63240	−0.314488
$590$	−14.0834	−0.579805
$591$	−4.92060	−0.202406
$592$	1.82419	0.0749738
$593$	34.2267	1.40552	0.702760	−	0.711427i	$-0.251951\pi$
$593$	34.2267	1.40552	0.702760	+	0.711427i	$0.251951\pi$
$594$	−2.32558	−0.0954199
$595$	0	0
$596$	13.5802	0.556265
$597$	13.4969	0.552392
$598$	0	0
$599$	14.9432	0.610564	0.305282	−	0.952262i	$-0.401249\pi$
$599$	14.9432	0.610564	0.305282	+	0.952262i	$0.401249\pi$
$600$	−5.11095	−0.208654
$601$	3.34458	0.136428	0.0682141	−	0.997671i	$-0.478270\pi$
$601$	3.34458	0.136428	0.0682141	+	0.997671i	$0.478270\pi$
$602$	0	0
$603$	23.1116	0.941178
$604$	−14.8344	−0.603603
$605$	−17.9593	−0.730151
$606$	−5.91279	−0.240191
$607$	6.10524	0.247804	0.123902	−	0.992294i	$-0.460459\pi$
$607$	6.10524	0.247804	0.123902	+	0.992294i	$0.460459\pi$
$608$	9.46235	0.383749
$609$	0	0
$610$	−6.58857	−0.266764
$611$	0	0
$612$	12.4476	0.503165
$613$	7.00014	0.282733	0.141366	−	0.989957i	$-0.454850\pi$
$613$	7.00014	0.282733	0.141366	+	0.989957i	$0.454850\pi$
$614$	30.0740	1.21369
$615$	−1.60159	−0.0645825
$616$	0	0
$617$	20.2468	0.815107	0.407553	−	0.913181i	$-0.366382\pi$
$617$	20.2468	0.815107	0.407553	+	0.913181i	$0.366382\pi$
$618$	−14.0232	−0.564094
$619$	38.2583	1.53773	0.768865	−	0.639411i	$-0.220822\pi$
$619$	38.2583	1.53773	0.768865	+	0.639411i	$0.220822\pi$
$620$	−5.83407	−0.234302
$621$	27.7757	1.11460
$622$	5.44604	0.218366
$623$	0	0
$624$	0	0
$625$	−9.26598	−0.370639
$626$	−16.3455	−0.653299
$627$	0.832871	0.0332617
$628$	2.67916	0.106910
$629$	−7.77864	−0.310155
$630$	0	0
$631$	−19.5070	−0.776561	−0.388280	−	0.921541i	$-0.626931\pi$
$631$	−19.5070	−0.776561	−0.388280	+	0.921541i	$0.626931\pi$
$632$	25.0877	0.997934
$633$	15.2298	0.605329
$634$	26.3955	1.04830
$635$	−17.8938	−0.710093
$636$	−9.51442	−0.377271
$637$	0	0
$638$	−2.72435	−0.107858
$639$	−3.66223	−0.144875
$640$	2.62903	0.103922
$641$	27.7916	1.09770	0.548851	−	0.835920i	$-0.315066\pi$
$641$	27.7916	1.09770	0.548851	+	0.835920i	$0.315066\pi$
$642$	−4.01930	−0.158629
$643$	43.2520	1.70569	0.852846	−	0.522162i	$-0.174874\pi$
$643$	43.2520	1.70569	0.852846	+	0.522162i	$0.174874\pi$
$644$	0	0
$645$	−4.21506	−0.165968
$646$	11.7925	0.463970
$647$	18.0100	0.708048	0.354024	−	0.935236i	$-0.384813\pi$
$647$	18.0100	0.708048	0.354024	+	0.935236i	$0.384813\pi$
$648$	12.2108	0.479684
$649$	−4.34026	−0.170370
$650$	0	0
$651$	0	0
$652$	15.4639	0.605615
$653$	−23.9251	−0.936261	−0.468130	−	0.883659i	$-0.655072\pi$
$653$	−23.9251	−0.936261	−0.468130	+	0.883659i	$0.655072\pi$
$654$	14.2958	0.559008
$655$	15.1143	0.590564
$656$	1.63891	0.0639887
$657$	−37.6747	−1.46983
$658$	0	0
$659$	−6.83720	−0.266340	−0.133170	−	0.991093i	$-0.542516\pi$
$659$	−6.83720	−0.266340	−0.133170	+	0.991093i	$0.542516\pi$
$660$	0.636631	0.0247808
$661$	−29.2000	−1.13575	−0.567874	−	0.823115i	$-0.692234\pi$
$661$	−29.2000	−1.13575	−0.567874	+	0.823115i	$0.692234\pi$
$662$	−14.8617	−0.577616
$663$	0	0
$664$	−28.6031	−1.11002
$665$	0	0
$666$	−3.46102	−0.134112
$667$	32.5384	1.25989
$668$	−22.8422	−0.883793
$669$	20.6602	0.798771
$670$	16.7363	0.646582
$671$	−2.03048	−0.0783858
$672$	0	0
$673$	−5.50832	−0.212330	−0.106165	−	0.994349i	$-0.533857\pi$
$673$	−5.50832	−0.212330	−0.106165	+	0.994349i	$0.533857\pi$
$674$	7.93787	0.305755
$675$	9.10674	0.350519
$676$	0	0
$677$	−9.80946	−0.377008	−0.188504	−	0.982072i	$-0.560364\pi$
$677$	−9.80946	−0.377008	−0.188504	+	0.982072i	$0.560364\pi$
$678$	6.06131	0.232783
$679$	0	0
$680$	28.6454	1.09850
$681$	1.26264	0.0483844
$682$	2.11778	0.0810940
$683$	9.90172	0.378879	0.189439	−	0.981892i	$-0.439333\pi$
$683$	9.90172	0.378879	0.189439	+	0.981892i	$0.439333\pi$
$684$	−4.45455	−0.170324
$685$	−6.92389	−0.264548
$686$	0	0
$687$	−5.37085	−0.204911
$688$	4.31326	0.164442
$689$	0	0
$690$	8.95620	0.340957
$691$	41.5146	1.57929	0.789646	−	0.613563i	$-0.210264\pi$
$691$	41.5146	1.57929	0.789646	+	0.613563i	$0.210264\pi$
$692$	0.142827	0.00542948
$693$	0	0
$694$	12.7601	0.484368
$695$	33.0501	1.25366
$696$	−11.3820	−0.431433
$697$	−6.98858	−0.264711
$698$	−17.4869	−0.661890
$699$	11.0495	0.417932
$700$	0	0
$701$	26.0138	0.982527	0.491263	−	0.871011i	$-0.336535\pi$
$701$	26.0138	0.982527	0.491263	+	0.871011i	$0.336535\pi$
$702$	0	0
$703$	2.78369	0.104989
$704$	−4.04435	−0.152427
$705$	4.90145	0.184599
$706$	11.5676	0.435353
$707$	0	0
$708$	−5.70602	−0.214445
$709$	33.5061	1.25835	0.629173	−	0.777265i	$-0.283394\pi$
$709$	33.5061	1.25835	0.629173	+	0.777265i	$0.283394\pi$
$710$	−2.65201	−0.0995282
$711$	−19.9044	−0.746474
$712$	6.49424	0.243382
$713$	−25.2938	−0.947261
$714$	0	0
$715$	0	0
$716$	−18.7563	−0.700958
$717$	−15.3843	−0.574539
$718$	35.7356	1.33364
$719$	26.4921	0.987988	0.493994	−	0.869465i	$-0.335536\pi$
$719$	26.4921	0.987988	0.493994	+	0.869465i	$0.335536\pi$
$720$	5.32979	0.198630
$721$	0	0
$722$	15.5406	0.578361
$723$	21.3127	0.792628
$724$	5.31504	0.197532
$725$	10.6682	0.396209
$726$	8.57072	0.318089
$727$	14.6388	0.542922	0.271461	−	0.962449i	$-0.412493\pi$
$727$	14.6388	0.542922	0.271461	+	0.962449i	$0.412493\pi$
$728$	0	0
$729$	−0.0845074	−0.00312990
$730$	−27.2822	−1.00976
$731$	−18.3925	−0.680269
$732$	−2.66942	−0.0986645
$733$	−24.3546	−0.899557	−0.449779	−	0.893140i	$-0.648497\pi$
$733$	−24.3546	−0.899557	−0.449779	+	0.893140i	$0.648497\pi$
$734$	−23.4524	−0.865643
$735$	0	0
$736$	31.3583	1.15588
$737$	5.15784	0.189992
$738$	−3.10949	−0.114462
$739$	−47.3631	−1.74228	−0.871140	−	0.491035i	$-0.836619\pi$
$739$	−47.3631	−1.74228	−0.871140	+	0.491035i	$0.836619\pi$
$740$	2.12780	0.0782197
$741$	0	0
$742$	0	0
$743$	22.9947	0.843593	0.421796	−	0.906691i	$-0.361400\pi$
$743$	22.9947	0.843593	0.421796	+	0.906691i	$0.361400\pi$
$744$	8.84783	0.324377
$745$	−24.7949	−0.908414
$746$	−12.6299	−0.462414
$747$	22.6936	0.830314
$748$	2.77795	0.101572
$749$	0	0
$750$	9.64459	0.352171
$751$	−13.2179	−0.482328	−0.241164	−	0.970484i	$-0.577529\pi$
$751$	−13.2179	−0.482328	−0.241164	+	0.970484i	$0.577529\pi$
$752$	−5.01565	−0.182902
$753$	−7.38253	−0.269034
$754$	0	0
$755$	27.0849	0.985721
$756$	0	0
$757$	0.817127	0.0296990	0.0148495	−	0.999890i	$-0.495273\pi$
$757$	0.817127	0.0296990	0.0148495	+	0.999890i	$0.495273\pi$
$758$	16.4230	0.596509
$759$	2.76014	0.100187
$760$	−10.2512	−0.371848
$761$	−43.5081	−1.57717	−0.788584	−	0.614927i	$-0.789186\pi$
$761$	−43.5081	−1.57717	−0.788584	+	0.614927i	$0.789186\pi$
$762$	8.53944	0.309351
$763$	0	0
$764$	7.76763	0.281023
$765$	−22.7271	−0.821700
$766$	−27.1004	−0.979176
$767$	0	0
$768$	−12.8345	−0.463124
$769$	19.6470	0.708490	0.354245	−	0.935153i	$-0.384738\pi$
$769$	19.6470	0.708490	0.354245	+	0.935153i	$0.384738\pi$
$770$	0	0
$771$	3.77961	0.136119
$772$	1.23664	0.0445077
$773$	−22.1970	−0.798372	−0.399186	−	0.916870i	$-0.630707\pi$
$773$	−22.1970	−0.798372	−0.399186	+	0.916870i	$0.630707\pi$
$774$	−8.18351	−0.294150
$775$	−8.29300	−0.297894
$776$	19.5601	0.702168
$777$	0	0
$778$	−13.4794	−0.483259
$779$	2.50096	0.0896062
$780$	0	0
$781$	−0.817302	−0.0292454
$782$	39.0805	1.39752
$783$	20.2805	0.724766
$784$	0	0
$785$	−4.89165	−0.174591
$786$	−7.21299	−0.257279
$787$	−8.63249	−0.307715	−0.153857	−	0.988093i	$-0.549170\pi$
$787$	−8.63249	−0.307715	−0.153857	+	0.988093i	$0.549170\pi$
$788$	5.87330	0.209228
$789$	14.2618	0.507735
$790$	−14.4138	−0.512822
$791$	0	0
$792$	3.92791	0.139572
$793$	0	0
$794$	−10.0161	−0.355460
$795$	17.3716	0.616107
$796$	−16.1101	−0.571008
$797$	39.5185	1.39982	0.699908	−	0.714233i	$-0.253224\pi$
$797$	39.5185	1.39982	0.699908	+	0.714233i	$0.253224\pi$
$798$	0	0
$799$	21.3875	0.756637
$800$	10.2813	0.363500
$801$	−5.15249	−0.182054
$802$	11.8272	0.417634
$803$	−8.40789	−0.296708
$804$	6.78087	0.239143
$805$	0	0
$806$	0	0
$807$	−10.6049	−0.373311
$808$	22.4282	0.789022
$809$	24.5166	0.861959	0.430979	−	0.902362i	$-0.358168\pi$
$809$	24.5166	0.861959	0.430979	+	0.902362i	$0.358168\pi$
$810$	−7.01556	−0.246502
$811$	19.0854	0.670179	0.335089	−	0.942186i	$-0.391233\pi$
$811$	19.0854	0.670179	0.335089	+	0.942186i	$0.391233\pi$
$812$	0	0
$813$	−15.3844	−0.539554
$814$	−0.772398	−0.0270726
$815$	−28.2343	−0.989005
$816$	−5.71660	−0.200121
$817$	6.58199	0.230275
$818$	22.5602	0.788800
$819$	0	0
$820$	1.91169	0.0667590
$821$	8.15752	0.284700	0.142350	−	0.989816i	$-0.454534\pi$
$821$	8.15752	0.284700	0.142350	+	0.989816i	$0.454534\pi$
$822$	3.30429	0.115250
$823$	48.5353	1.69183	0.845917	−	0.533315i	$-0.179054\pi$
$823$	48.5353	1.69183	0.845917	+	0.533315i	$0.179054\pi$
$824$	53.1922	1.85304
$825$	0.904958	0.0315066
$826$	0	0
$827$	−2.45879	−0.0855004	−0.0427502	−	0.999086i	$-0.513612\pi$
$827$	−2.45879	−0.0855004	−0.0427502	+	0.999086i	$0.513612\pi$
$828$	−14.7624	−0.513029
$829$	−36.0195	−1.25101	−0.625504	−	0.780221i	$-0.715107\pi$
$829$	−36.0195	−1.25101	−0.625504	+	0.780221i	$0.715107\pi$
$830$	16.4336	0.570419
$831$	19.9124	0.690754
$832$	0	0
$833$	0	0
$834$	−15.7725	−0.546156
$835$	41.7058	1.44329
$836$	−0.994127	−0.0343826
$837$	−15.7651	−0.544923
$838$	6.15279	0.212545
$839$	33.2440	1.14771	0.573855	−	0.818957i	$-0.305447\pi$
$839$	33.2440	1.14771	0.573855	+	0.818957i	$0.305447\pi$
$840$	0	0
$841$	−5.24203	−0.180760
$842$	1.44672	0.0498573
$843$	15.5742	0.536404
$844$	−18.1785	−0.625729
$845$	0	0
$846$	9.51614	0.327172
$847$	0	0
$848$	−17.7763	−0.610442
$849$	17.2302	0.591339
$850$	12.8132	0.439489
$851$	9.22518	0.316235
$852$	−1.07449	−0.0368113
$853$	3.02993	0.103743	0.0518714	−	0.998654i	$-0.483481\pi$
$853$	3.02993	0.103743	0.0518714	+	0.998654i	$0.483481\pi$
$854$	0	0
$855$	8.13320	0.278150
$856$	15.2459	0.521094
$857$	35.8756	1.22549	0.612744	−	0.790282i	$-0.290066\pi$
$857$	35.8756	1.22549	0.612744	+	0.790282i	$0.290066\pi$
$858$	0	0
$859$	−16.1467	−0.550918	−0.275459	−	0.961313i	$-0.588830\pi$
$859$	−16.1467	−0.550918	−0.275459	+	0.961313i	$0.588830\pi$
$860$	5.03115	0.171561
$861$	0	0
$862$	−5.23840	−0.178420
$863$	8.12050	0.276425	0.138212	−	0.990403i	$-0.455864\pi$
$863$	8.12050	0.276425	0.138212	+	0.990403i	$0.455864\pi$
$864$	19.5450	0.664934
$865$	−0.260777	−0.00886667
$866$	15.3311	0.520972
$867$	11.2973	0.383677
$868$	0	0
$869$	−4.44209	−0.150687
$870$	6.53939	0.221706
$871$	0	0
$872$	−54.2262	−1.83633
$873$	−15.5189	−0.525235
$874$	−13.9855	−0.473066
$875$	0	0
$876$	−11.0536	−0.373467
$877$	21.8404	0.737498	0.368749	−	0.929529i	$-0.379786\pi$
$877$	21.8404	0.737498	0.368749	+	0.929529i	$0.379786\pi$
$878$	7.96488	0.268802
$879$	−17.6356	−0.594833
$880$	1.18945	0.0400965
$881$	−40.8700	−1.37695	−0.688473	−	0.725262i	$-0.741718\pi$
$881$	−40.8700	−1.37695	−0.688473	+	0.725262i	$0.741718\pi$
$882$	0	0
$883$	−7.78501	−0.261986	−0.130993	−	0.991383i	$-0.541817\pi$
$883$	−7.78501	−0.261986	−0.130993	+	0.991383i	$0.541817\pi$
$884$	0	0
$885$	10.4181	0.350202
$886$	−31.2604	−1.05021
$887$	0.482171	0.0161897	0.00809486	−	0.999967i	$-0.497423\pi$
$887$	0.482171	0.0161897	0.00809486	+	0.999967i	$0.497423\pi$
$888$	−3.22699	−0.108291
$889$	0	0
$890$	−3.73119	−0.125070
$891$	−2.16207	−0.0724320
$892$	−24.6604	−0.825690
$893$	−7.65383	−0.256125
$894$	11.8328	0.395750
$895$	34.2457	1.14471
$896$	0	0
$897$	0	0
$898$	−34.4526	−1.14970
$899$	−18.4683	−0.615954
$900$	−4.84011	−0.161337
$901$	75.8012	2.52530
$902$	−0.693947	−0.0231059
$903$	0	0
$904$	−22.9916	−0.764688
$905$	−9.70429	−0.322582
$906$	−12.9257	−0.429428
$907$	5.90300	0.196006	0.0980030	−	0.995186i	$-0.468755\pi$
$907$	5.90300	0.196006	0.0980030	+	0.995186i	$0.468755\pi$
$908$	−1.50710	−0.0500150
$909$	−17.7944	−0.590203
$910$	0	0
$911$	−39.3536	−1.30384	−0.651921	−	0.758287i	$-0.726037\pi$
$911$	−39.3536	−1.30384	−0.651921	+	0.758287i	$0.726037\pi$
$912$	2.04576	0.0677420
$913$	5.06454	0.167612
$914$	−22.9368	−0.758682
$915$	4.87387	0.161125
$916$	6.41072	0.211816
$917$	0	0
$918$	24.3581	0.803937
$919$	−18.4516	−0.608663	−0.304332	−	0.952566i	$-0.598433\pi$
$919$	−18.4516	−0.608663	−0.304332	+	0.952566i	$0.598433\pi$
$920$	−33.9724	−1.12004
$921$	−22.2471	−0.733067
$922$	−22.9005	−0.754187
$923$	0	0
$924$	0	0
$925$	3.02463	0.0994492
$926$	−36.4371	−1.19740
$927$	−42.2024	−1.38611
$928$	22.8963	0.751609
$929$	3.47157	0.113899	0.0569493	−	0.998377i	$-0.481863\pi$
$929$	3.47157	0.113899	0.0569493	+	0.998377i	$0.481863\pi$
$930$	−5.08342	−0.166692
$931$	0	0
$932$	−13.1889	−0.432017
$933$	−4.02868	−0.131893
$934$	19.7772	0.647129
$935$	−5.07203	−0.165873
$936$	0	0
$937$	16.7987	0.548789	0.274395	−	0.961617i	$-0.411523\pi$
$937$	16.7987	0.548789	0.274395	+	0.961617i	$0.411523\pi$
$938$	0	0
$939$	12.0915	0.394592
$940$	−5.85044	−0.190820
$941$	1.56115	0.0508922	0.0254461	−	0.999676i	$-0.491899\pi$
$941$	1.56115	0.0508922	0.0254461	+	0.999676i	$0.491899\pi$
$942$	2.33444	0.0760602
$943$	8.28819	0.269901
$944$	−10.6609	−0.346982
$945$	0	0
$946$	−1.82632	−0.0593788
$947$	44.1853	1.43583	0.717915	−	0.696130i	$-0.245097\pi$
$947$	44.1853	1.43583	0.717915	+	0.696130i	$0.245097\pi$
$948$	−5.83989	−0.189671
$949$	0	0
$950$	−4.58538	−0.148769
$951$	−19.5260	−0.633173
$952$	0	0
$953$	−14.3509	−0.464873	−0.232436	−	0.972612i	$-0.574670\pi$
$953$	−14.3509	−0.464873	−0.232436	+	0.972612i	$0.574670\pi$
$954$	33.7269	1.09195
$955$	−14.1823	−0.458928
$956$	18.3630	0.593901
$957$	2.01532	0.0651461
$958$	−21.1231	−0.682458
$959$	0	0
$960$	9.70788	0.313320
$961$	−16.6436	−0.536889
$962$	0	0
$963$	−12.0960	−0.389788
$964$	−25.4392	−0.819340
$965$	−2.25788	−0.0726838
$966$	0	0
$967$	−46.1818	−1.48511	−0.742553	−	0.669787i	$-0.766385\pi$
$967$	−46.1818	−1.48511	−0.742553	+	0.669787i	$0.766385\pi$
$968$	−32.5102	−1.04492
$969$	−8.72346	−0.280238
$970$	−11.2381	−0.360832
$971$	−41.9325	−1.34568	−0.672839	−	0.739789i	$-0.734925\pi$
$971$	−41.9325	−1.34568	−0.672839	+	0.739789i	$0.734925\pi$
$972$	−14.3052	−0.458840
$973$	0	0
$974$	6.69231	0.214436
$975$	0	0
$976$	−4.98742	−0.159644
$977$	3.14579	0.100643	0.0503214	−	0.998733i	$-0.483975\pi$
$977$	3.14579	0.100643	0.0503214	+	0.998733i	$0.483975\pi$
$978$	13.4743	0.430859
$979$	−1.14989	−0.0367505
$980$	0	0
$981$	43.0227	1.37361
$982$	−0.998879	−0.0318755
$983$	−49.4907	−1.57851	−0.789254	−	0.614067i	$-0.789533\pi$
$983$	−49.4907	−1.57851	−0.789254	+	0.614067i	$0.789533\pi$
$984$	−2.89923	−0.0924240
$985$	−10.7236	−0.341681
$986$	28.5347	0.908730
$987$	0	0
$988$	0	0
$989$	21.8128	0.693606
$990$	−2.25674	−0.0717239
$991$	16.5615	0.526094	0.263047	−	0.964783i	$-0.415273\pi$
$991$	16.5615	0.526094	0.263047	+	0.964783i	$0.415273\pi$
$992$	−17.7985	−0.565105
$993$	10.9939	0.348880
$994$	0	0
$995$	29.4141	0.932491
$996$	6.65822	0.210974
$997$	−24.3432	−0.770958	−0.385479	−	0.922717i	$-0.625964\pi$
$997$	−24.3432	−0.770958	−0.385479	+	0.922717i	$0.625964\pi$
$998$	−10.1804	−0.322254
$999$	5.74987	0.181918

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.cx.1.12		32
7.6	odd	2	inner	8281.2.a.cx.1.11		32
13.6	odd	12		637.2.q.j.491.11	✓	32
13.11	odd	12		637.2.q.j.589.11	yes	32
13.12	even	2	inner	8281.2.a.cx.1.22		32
91.6	even	12		637.2.q.j.491.12	yes	32
91.11	odd	12		637.2.u.j.30.6		32
91.19	even	12		637.2.u.j.361.5		32
91.24	even	12		637.2.u.j.30.5		32
91.32	odd	12		637.2.k.j.569.11		32
91.37	odd	12		637.2.k.j.459.5		32
91.45	even	12		637.2.k.j.569.12		32
91.58	odd	12		637.2.u.j.361.6		32
91.76	even	12		637.2.q.j.589.12	yes	32
91.89	even	12		637.2.k.j.459.6		32
91.90	odd	2	inner	8281.2.a.cx.1.21		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.k.j.459.5		32	91.37	odd	12
637.2.k.j.459.6		32	91.89	even	12
637.2.k.j.569.11		32	91.32	odd	12
637.2.k.j.569.12		32	91.45	even	12
637.2.q.j.491.11	✓	32	13.6	odd	12
637.2.q.j.491.12	yes	32	91.6	even	12
637.2.q.j.589.11	yes	32	13.11	odd	12
637.2.q.j.589.12	yes	32	91.76	even	12
637.2.u.j.30.5		32	91.24	even	12
637.2.u.j.30.6		32	91.11	odd	12
637.2.u.j.361.5		32	91.19	even	12
637.2.u.j.361.6		32	91.58	odd	12
8281.2.a.cx.1.11		32	7.6	odd	2	inner
8281.2.a.cx.1.12		32	1.1	even	1	trivial
8281.2.a.cx.1.21		32	91.90	odd	2	inner
8281.2.a.cx.1.22		32	13.12	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q-1.04004 q^{2} +0.769363 q^{3} -0.918323 q^{4} +1.67669 q^{5} -0.800166 q^{6} +3.03516 q^{8} -2.40808 q^{9} +O(q^{10})\)	\(q-1.04004 q^{2} +0.769363 q^{3} -0.918323 q^{4} +1.67669 q^{5} -0.800166 q^{6} +3.03516 q^{8} -2.40808 q^{9} -1.74382 q^{10} -0.537414 q^{11} -0.706523 q^{12} +1.28998 q^{15} -1.32004 q^{16} +5.62885 q^{17} +2.50449 q^{18} -2.01436 q^{19} -1.53974 q^{20} +0.558930 q^{22} -6.67561 q^{23} +2.33514 q^{24} -2.18871 q^{25} -4.16078 q^{27} -4.87421 q^{29} -1.34163 q^{30} +3.78899 q^{31} -4.69744 q^{32} -0.413466 q^{33} -5.85422 q^{34} +2.21140 q^{36} -1.38192 q^{37} +2.09501 q^{38} +5.08903 q^{40} -1.24156 q^{41} -3.26753 q^{43} +0.493519 q^{44} -4.03760 q^{45} +6.94288 q^{46} +3.79963 q^{47} -1.01559 q^{48} +2.27634 q^{50} +4.33063 q^{51} +13.4665 q^{53} +4.32736 q^{54} -0.901076 q^{55} -1.54978 q^{57} +5.06936 q^{58} +8.07619 q^{59} -1.18462 q^{60} +3.77824 q^{61} -3.94069 q^{62} +7.52559 q^{64} +0.430020 q^{66} -9.59752 q^{67} -5.16910 q^{68} -5.13597 q^{69} +1.52081 q^{71} -7.30892 q^{72} +15.6451 q^{73} +1.43725 q^{74} -1.68391 q^{75} +1.84984 q^{76} +8.26567 q^{79} -2.21329 q^{80} +4.02310 q^{81} +1.29127 q^{82} -9.42392 q^{83} +9.43784 q^{85} +3.39835 q^{86} -3.75004 q^{87} -1.63114 q^{88} +2.13967 q^{89} +4.19926 q^{90} +6.13037 q^{92} +2.91511 q^{93} -3.95175 q^{94} -3.37746 q^{95} -3.61403 q^{96} +6.44451 q^{97} +1.29414 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 40 q^{4} + 56 q^{9}+O(q^{10}) \) 32 * q + 40 * q^4 + 56 * q^9	\( 32 q + 40 q^{4} + 56 q^{9} + 56 q^{16} - 16 q^{22} + 48 q^{23} + 40 q^{25} + 48 q^{29} + 48 q^{30} + 184 q^{36} + 24 q^{43} + 72 q^{53} - 32 q^{64} - 48 q^{74} + 96 q^{79} + 128 q^{81} + 112 q^{88} + 168 q^{92} + 168 q^{95}+O(q^{100}) \) 32 * q + 40 * q^4 + 56 * q^9 + 56 * q^16 - 16 * q^22 + 48 * q^23 + 40 * q^25 + 48 * q^29 + 48 * q^30 + 184 * q^36 + 24 * q^43 + 72 * q^53 - 32 * q^64 - 48 * q^74 + 96 * q^79 + 128 * q^81 + 112 * q^88 + 168 * q^92 + 168 * q^95

Introduction

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