LMFDB - Embedded newform 8281.2.a.ce.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8281,2,Mod(1,8281)] 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("8281.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [6,2,-1,4,1,-9,0,3,-3,-4,4,-5,0,0,-2,-8,-5] 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:	$66.1241179138$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.6995813.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 6x^{4} + 4x^{3} + 7x^{2} - x - 1 $ x^6 - x^5 - 6x^4 + 4x^3 + 7*x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.51235$ of defining polynomial
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)

$f(q)$	$=$	\(q+0.851125 q^{2} +0.661223 q^{3} -1.27559 q^{4} -3.44148 q^{5} +0.562784 q^{6} -2.78793 q^{8} -2.56278 q^{9} -2.92913 q^{10} +0.897986 q^{11} -0.843447 q^{12} -2.27559 q^{15} +0.178289 q^{16} +1.93681 q^{17} -2.18125 q^{18} -1.03804 q^{19} +4.38990 q^{20} +0.764299 q^{22} +5.65013 q^{23} -1.84345 q^{24} +6.84378 q^{25} -3.67824 q^{27} -1.83594 q^{29} -1.93681 q^{30} +9.13385 q^{31} +5.72761 q^{32} +0.593769 q^{33} +1.64847 q^{34} +3.26905 q^{36} +10.6000 q^{37} -0.883501 q^{38} +9.59462 q^{40} +5.33143 q^{41} -3.91465 q^{43} -1.14546 q^{44} +8.81977 q^{45} +4.80897 q^{46} -7.19129 q^{47} +0.117889 q^{48} +5.82491 q^{50} +1.28066 q^{51} -9.38648 q^{53} -3.13065 q^{54} -3.09040 q^{55} -0.686375 q^{57} -1.56261 q^{58} +0.510517 q^{59} +2.90270 q^{60} +1.43619 q^{61} +7.77405 q^{62} +4.51834 q^{64} +0.505372 q^{66} +8.44932 q^{67} -2.47057 q^{68} +3.73600 q^{69} +3.44837 q^{71} +7.14487 q^{72} -10.9005 q^{73} +9.02195 q^{74} +4.52526 q^{75} +1.32411 q^{76} -12.0918 q^{79} -0.613579 q^{80} +5.25621 q^{81} +4.53771 q^{82} -1.51669 q^{83} -6.66549 q^{85} -3.33185 q^{86} -1.21396 q^{87} -2.50353 q^{88} -13.6078 q^{89} +7.50673 q^{90} -7.20722 q^{92} +6.03951 q^{93} -6.12069 q^{94} +3.57239 q^{95} +3.78723 q^{96} -0.506241 q^{97} -2.30134 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} - q^{3} + 4 q^{4} + q^{5} - 9 q^{6} + 3 q^{8} - 3 q^{9} - 4 q^{10} + 4 q^{11} - 5 q^{12} - 2 q^{15} - 8 q^{16} - 5 q^{17} + 3 q^{18} - q^{19} - q^{20} + 5 q^{22} + q^{23} - 11 q^{24}+ \cdots + 10 q^{99}+O(q^{100}) $ 6 * q + 2 * q^2 - q^3 + 4 * q^4 + q^5 - 9 * q^6 + 3 * q^8 - 3 * q^9 - 4 * q^10 + 4 * q^11 - 5 * q^12 - 2 * q^15 - 8 * q^16 - 5 * q^17 + 3 * q^18 - q^19 - q^20 + 5 * q^22 + q^23 - 11 * q^24 - 7 * q^25 - 4 * q^27 - 3 * q^29 + 5 * q^30 + 16 * q^31 + 8 * q^32 + 16 * q^33 - 16 * q^34 + 21 * q^36 - 13 * q^37 + 17 * q^38 + 5 * q^40 - 8 * q^41 + 11 * q^43 + 21 * q^44 - 7 * q^45 + 16 * q^46 - q^47 - 21 * q^48 + 6 * q^50 + 20 * q^51 + 2 * q^53 - 18 * q^54 - 9 * q^55 - 21 * q^57 - 8 * q^58 + 13 * q^59 + 20 * q^60 + 5 * q^61 - 5 * q^62 - 15 * q^64 - 18 * q^66 - 11 * q^67 - 29 * q^68 - 23 * q^69 + 6 * q^71 + 25 * q^72 - 30 * q^73 + 3 * q^74 + 3 * q^75 - 9 * q^76 - 7 * q^79 - 7 * q^80 + 6 * q^81 - q^82 + 27 * q^83 - q^85 - 7 * q^86 - 16 * q^87 + 4 * q^89 - 8 * q^90 + 27 * q^92 - 7 * q^93 - 45 * q^94 + 6 * q^95 + 19 * q^96 - 35 * q^97 + 10 * 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 $

$2$	0.851125	0.601837	0.300918	−	0.953650i	$-0.402707\pi$
$2$	0.851125	0.601837	0.300918	+	0.953650i	$0.402707\pi$
$3$	0.661223	0.381757	0.190879	−	0.981614i	$-0.438866\pi$
$3$	0.661223	0.381757	0.190879	+	0.981614i	$0.438866\pi$
$4$	−1.27559	−0.637793
$5$	−3.44148	−1.53908	−0.769538	−	0.638601i	$-0.779513\pi$
$5$	−3.44148	−1.53908	−0.769538	+	0.638601i	$0.779513\pi$
$6$	0.562784	0.229756
$7$	0	0
$7$	0	0
$8$	−2.78793	−0.985684
$9$	−2.56278	−0.854261
$10$	−2.92913	−0.926272
$11$	0.897986	0.270753	0.135377	−	0.990794i	$-0.456776\pi$
$11$	0.897986	0.270753	0.135377	+	0.990794i	$0.456776\pi$
$12$	−0.843447	−0.243482
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−2.27559	−0.587554
$16$	0.178289	0.0445723
$17$	1.93681	0.469745	0.234873	−	0.972026i	$-0.424533\pi$
$17$	1.93681	0.469745	0.234873	+	0.972026i	$0.424533\pi$
$18$	−2.18125	−0.514126
$19$	−1.03804	−0.238142	−0.119071	−	0.992886i	$-0.537992\pi$
$19$	−1.03804	−0.238142	−0.119071	+	0.992886i	$0.537992\pi$
$20$	4.38990	0.981612
$21$	0	0
$22$	0.764299	0.162949
$23$	5.65013	1.17813	0.589067	−	0.808084i	$-0.299496\pi$
$23$	5.65013	1.17813	0.589067	+	0.808084i	$0.299496\pi$
$24$	−1.84345	−0.376292
$25$	6.84378	1.36876
$26$	0	0
$27$	−3.67824	−0.707878
$28$	0	0
$29$	−1.83594	−0.340925	−0.170463	−	0.985364i	$-0.554526\pi$
$29$	−1.83594	−0.340925	−0.170463	+	0.985364i	$0.554526\pi$
$30$	−1.93681	−0.353611
$31$	9.13385	1.64049	0.820244	−	0.572014i	$-0.193838\pi$
$31$	9.13385	1.64049	0.820244	+	0.572014i	$0.193838\pi$
$32$	5.72761	1.01251
$33$	0.593769	0.103362
$34$	1.64847	0.282710
$35$	0	0
$36$	3.26905	0.544842
$37$	10.6000	1.74263	0.871316	−	0.490722i	$-0.163267\pi$
$37$	10.6000	1.74263	0.871316	+	0.490722i	$0.163267\pi$
$38$	−0.883501	−0.143323
$39$	0	0
$40$	9.59462	1.51704
$41$	5.33143	0.832629	0.416314	−	0.909221i	$-0.363322\pi$
$41$	5.33143	0.832629	0.416314	+	0.909221i	$0.363322\pi$
$42$	0	0
$43$	−3.91465	−0.596978	−0.298489	−	0.954413i	$-0.596483\pi$
$43$	−3.91465	−0.596978	−0.298489	+	0.954413i	$0.596483\pi$
$44$	−1.14546	−0.172684
$45$	8.81977	1.31477
$46$	4.80897	0.709044
$47$	−7.19129	−1.04896	−0.524479	−	0.851423i	$-0.675740\pi$
$47$	−7.19129	−1.04896	−0.524479	+	0.851423i	$0.675740\pi$
$48$	0.117889	0.0170158
$49$	0	0
$50$	5.82491	0.823767
$51$	1.28066	0.179329
$52$	0	0
$53$	−9.38648	−1.28933	−0.644666	−	0.764464i	$-0.723004\pi$
$53$	−9.38648	−1.28933	−0.644666	+	0.764464i	$0.723004\pi$
$54$	−3.13065	−0.426027
$55$	−3.09040	−0.416710
$56$	0	0
$57$	−0.686375	−0.0909127
$58$	−1.56261	−0.205181
$59$	0.510517	0.0664637	0.0332318	−	0.999448i	$-0.489420\pi$
$59$	0.510517	0.0664637	0.0332318	+	0.999448i	$0.489420\pi$
$60$	2.90270	0.374737
$61$	1.43619	0.183885	0.0919426	−	0.995764i	$-0.470692\pi$
$61$	1.43619	0.183885	0.0919426	+	0.995764i	$0.470692\pi$
$62$	7.77405	0.987305
$63$	0	0
$64$	4.51834	0.564792
$65$	0	0
$66$	0.505372	0.0622070
$67$	8.44932	1.03225	0.516124	−	0.856514i	$-0.327374\pi$
$67$	8.44932	1.03225	0.516124	+	0.856514i	$0.327374\pi$
$68$	−2.47057	−0.299600
$69$	3.73600	0.449761
$70$	0	0
$71$	3.44837	0.409247	0.204623	−	0.978841i	$-0.434403\pi$
$71$	3.44837	0.409247	0.204623	+	0.978841i	$0.434403\pi$
$72$	7.14487	0.842031
$73$	−10.9005	−1.27581	−0.637905	−	0.770115i	$-0.720199\pi$
$73$	−10.9005	−1.27581	−0.637905	+	0.770115i	$0.720199\pi$
$74$	9.02195	1.04878
$75$	4.52526	0.522532
$76$	1.32411	0.151886
$77$	0	0
$78$	0	0
$79$	−12.0918	−1.36043	−0.680216	−	0.733012i	$-0.738114\pi$
$79$	−12.0918	−1.36043	−0.680216	+	0.733012i	$0.738114\pi$
$80$	−0.613579	−0.0686002
$81$	5.25621	0.584024
$82$	4.53771	0.501107
$83$	−1.51669	−0.166479	−0.0832393	−	0.996530i	$-0.526527\pi$
$83$	−1.51669	−0.166479	−0.0832393	+	0.996530i	$0.526527\pi$
$84$	0	0
$85$	−6.66549	−0.722973
$86$	−3.33185	−0.359283
$87$	−1.21396	−0.130151
$88$	−2.50353	−0.266877
$89$	−13.6078	−1.44243	−0.721213	−	0.692714i	$-0.756415\pi$
$89$	−13.6078	−1.44243	−0.721213	+	0.692714i	$0.756415\pi$
$90$	7.50673	0.791279
$91$	0	0
$92$	−7.20722	−0.751405
$93$	6.03951	0.626268
$94$	−6.12069	−0.631301
$95$	3.57239	0.366519
$96$	3.78723	0.386533
$97$	−0.506241	−0.0514010	−0.0257005	−	0.999670i	$-0.508182\pi$
$97$	−0.506241	−0.0514010	−0.0257005	+	0.999670i	$0.508182\pi$
$98$	0	0
$99$	−2.30134	−0.231294
$100$	−8.72982	−0.872982
$101$	−5.98654	−0.595683	−0.297842	−	0.954615i	$-0.596267\pi$
$101$	−5.98654	−0.595683	−0.297842	+	0.954615i	$0.596267\pi$
$102$	1.09000	0.107927
$103$	−4.13302	−0.407239	−0.203619	−	0.979050i	$-0.565270\pi$
$103$	−4.13302	−0.407239	−0.203619	+	0.979050i	$0.565270\pi$
$104$	0	0
$105$	0	0
$106$	−7.98908	−0.775968
$107$	−14.1234	−1.36536	−0.682679	−	0.730718i	$-0.739185\pi$
$107$	−14.1234	−1.36536	−0.682679	+	0.730718i	$0.739185\pi$
$108$	4.69191	0.451479
$109$	4.20237	0.402514	0.201257	−	0.979538i	$-0.435497\pi$
$109$	4.20237	0.402514	0.201257	+	0.979538i	$0.435497\pi$
$110$	−2.63032	−0.250791
$111$	7.00898	0.665263
$112$	0	0
$113$	13.7694	1.29532	0.647660	−	0.761929i	$-0.275748\pi$
$113$	13.7694	1.29532	0.647660	+	0.761929i	$0.275748\pi$
$114$	−0.584192	−0.0547146
$115$	−19.4448	−1.81324
$116$	2.34190	0.217440
$117$	0	0
$118$	0.434514	0.0400003
$119$	0	0
$120$	6.34418	0.579142
$121$	−10.1936	−0.926693
$122$	1.22238	0.110669
$123$	3.52526	0.317862
$124$	−11.6510	−1.04629
$125$	−6.34531	−0.567542
$126$	0	0
$127$	1.94496	0.172588	0.0862938	−	0.996270i	$-0.472498\pi$
$127$	1.94496	0.172588	0.0862938	+	0.996270i	$0.472498\pi$
$128$	−7.60956	−0.672596
$129$	−2.58845	−0.227901
$130$	0	0
$131$	−12.0354	−1.05154	−0.525769	−	0.850627i	$-0.676222\pi$
$131$	−12.0354	−1.05154	−0.525769	+	0.850627i	$0.676222\pi$
$132$	−0.757404	−0.0659235
$133$	0	0
$134$	7.19143	0.621245
$135$	12.6586	1.08948
$136$	−5.39970	−0.463020
$137$	−8.71715	−0.744756	−0.372378	−	0.928081i	$-0.621457\pi$
$137$	−8.71715	−0.744756	−0.372378	+	0.928081i	$0.621457\pi$
$138$	3.17980	0.270683
$139$	4.21250	0.357299	0.178650	−	0.983913i	$-0.442827\pi$
$139$	4.21250	0.357299	0.178650	+	0.983913i	$0.442827\pi$
$140$	0	0
$141$	−4.75505	−0.400447
$142$	2.93500	0.246300
$143$	0	0
$144$	−0.456917	−0.0380764
$145$	6.31834	0.524710
$146$	−9.27771	−0.767829
$147$	0	0
$148$	−13.5212	−1.11144
$149$	−5.86484	−0.480466	−0.240233	−	0.970715i	$-0.577224\pi$
$149$	−5.86484	−0.480466	−0.240233	+	0.970715i	$0.577224\pi$
$150$	3.85157	0.314479
$151$	16.8568	1.37179	0.685893	−	0.727702i	$-0.259412\pi$
$151$	16.8568	1.37179	0.685893	+	0.727702i	$0.259412\pi$
$152$	2.89398	0.234733
$153$	−4.96362	−0.401285
$154$	0	0
$155$	−31.4339	−2.52483
$156$	0	0
$157$	−1.93900	−0.154749	−0.0773746	−	0.997002i	$-0.524654\pi$
$157$	−1.93900	−0.154749	−0.0773746	+	0.997002i	$0.524654\pi$
$158$	−10.2916	−0.818757
$159$	−6.20656	−0.492212
$160$	−19.7115	−1.55833
$161$	0	0
$162$	4.47370	0.351487
$163$	11.8959	0.931762	0.465881	−	0.884847i	$-0.345738\pi$
$163$	11.8959	0.931762	0.465881	+	0.884847i	$0.345738\pi$
$164$	−6.80069	−0.531045
$165$	−2.04344	−0.159082
$166$	−1.29090	−0.100193
$167$	−16.5760	−1.28269	−0.641346	−	0.767252i	$-0.721624\pi$
$167$	−16.5760	−1.28269	−0.641346	+	0.767252i	$0.721624\pi$
$168$	0	0
$169$	0	0
$170$	−5.67316	−0.435112
$171$	2.66027	0.203436
$172$	4.99346	0.380748
$173$	−9.98656	−0.759264	−0.379632	−	0.925138i	$-0.623949\pi$
$173$	−9.98656	−0.759264	−0.379632	+	0.925138i	$0.623949\pi$
$174$	−1.03324	−0.0783295
$175$	0	0
$176$	0.160101	0.0120681
$177$	0.337566	0.0253730
$178$	−11.5820	−0.868105
$179$	9.17657	0.685889	0.342945	−	0.939356i	$-0.388576\pi$
$179$	9.17657	0.685889	0.342945	+	0.939356i	$0.388576\pi$
$180$	−11.2504	−0.838553
$181$	6.00489	0.446340	0.223170	−	0.974780i	$-0.428360\pi$
$181$	6.00489	0.446340	0.223170	+	0.974780i	$0.428360\pi$
$182$	0	0
$183$	0.949642	0.0701995
$184$	−15.7522	−1.16127
$185$	−36.4797	−2.68204
$186$	5.14038	0.376911
$187$	1.73923	0.127185
$188$	9.17311	0.669018
$189$	0	0
$190$	3.04055	0.220585
$191$	1.31612	0.0952313	0.0476156	−	0.998866i	$-0.484838\pi$
$191$	1.31612	0.0952313	0.0476156	+	0.998866i	$0.484838\pi$
$192$	2.98763	0.215614
$193$	16.4254	1.18233	0.591163	−	0.806552i	$-0.298669\pi$
$193$	16.4254	1.18233	0.591163	+	0.806552i	$0.298669\pi$
$194$	−0.430874	−0.0309350
$195$	0	0
$196$	0	0
$197$	25.5875	1.82303	0.911517	−	0.411262i	$-0.134912\pi$
$197$	25.5875	1.82303	0.911517	+	0.411262i	$0.134912\pi$
$198$	−1.95873	−0.139201
$199$	−25.3788	−1.79906	−0.899528	−	0.436864i	$-0.856089\pi$
$199$	−25.3788	−1.79906	−0.899528	+	0.436864i	$0.856089\pi$
$200$	−19.0800	−1.34916
$201$	5.58688	0.394068
$202$	−5.09530	−0.358504
$203$	0	0
$204$	−1.63359	−0.114375
$205$	−18.3480	−1.28148
$206$	−3.51772	−0.245091
$207$	−14.4801	−1.00643
$208$	0	0
$209$	−0.932145	−0.0644778
$210$	0	0
$211$	−5.69648	−0.392162	−0.196081	−	0.980588i	$-0.562822\pi$
$211$	−5.69648	−0.392162	−0.196081	+	0.980588i	$0.562822\pi$
$212$	11.9733	0.822327
$213$	2.28014	0.156233
$214$	−12.0208	−0.821723
$215$	13.4722	0.918794
$216$	10.2547	0.697744
$217$	0	0
$218$	3.57675	0.242248
$219$	−7.20768	−0.487050
$220$	3.94207	0.265774
$221$	0	0
$222$	5.96552	0.400380
$223$	−2.35812	−0.157912	−0.0789558	−	0.996878i	$-0.525159\pi$
$223$	−2.35812	−0.157912	−0.0789558	+	0.996878i	$0.525159\pi$
$224$	0	0
$225$	−17.5391	−1.16927
$226$	11.7195	0.779571
$227$	−26.2926	−1.74510	−0.872551	−	0.488523i	$-0.837536\pi$
$227$	−26.2926	−1.74510	−0.872551	+	0.488523i	$0.837536\pi$
$228$	0.875531	0.0579834
$229$	−0.0685555	−0.00453027	−0.00226514	−	0.999997i	$-0.500721\pi$
$229$	−0.0685555	−0.00453027	−0.00226514	+	0.999997i	$0.500721\pi$
$230$	−16.5500	−1.09127
$231$	0	0
$232$	5.11847	0.336044
$233$	14.6703	0.961082	0.480541	−	0.876972i	$-0.340440\pi$
$233$	14.6703	0.961082	0.480541	+	0.876972i	$0.340440\pi$
$234$	0	0
$235$	24.7487	1.61443
$236$	−0.651208	−0.0423901
$237$	−7.99536	−0.519355
$238$	0	0
$239$	−3.35434	−0.216974	−0.108487	−	0.994098i	$-0.534601\pi$
$239$	−3.35434	−0.216974	−0.108487	+	0.994098i	$0.534601\pi$
$240$	−0.405713	−0.0261886
$241$	8.57978	0.552672	0.276336	−	0.961061i	$-0.410880\pi$
$241$	8.57978	0.552672	0.276336	+	0.961061i	$0.410880\pi$
$242$	−8.67605	−0.557718
$243$	14.5103	0.930833
$244$	−1.83198	−0.117281
$245$	0	0
$246$	3.00044	0.191301
$247$	0	0
$248$	−25.4646	−1.61700
$249$	−1.00287	−0.0635544
$250$	−5.40066	−0.341568
$251$	21.5151	1.35802	0.679010	−	0.734129i	$-0.262409\pi$
$251$	21.5151	1.35802	0.679010	+	0.734129i	$0.262409\pi$
$252$	0	0
$253$	5.07374	0.318983
$254$	1.65541	0.103870
$255$	−4.40737	−0.276000
$256$	−15.5134	−0.969585
$257$	4.93792	0.308019	0.154010	−	0.988069i	$-0.450781\pi$
$257$	4.93792	0.308019	0.154010	+	0.988069i	$0.450781\pi$
$258$	−2.20310	−0.137159
$259$	0	0
$260$	0	0
$261$	4.70511	0.291239
$262$	−10.2436	−0.632854
$263$	−8.95439	−0.552151	−0.276076	−	0.961136i	$-0.589034\pi$
$263$	−8.95439	−0.552151	−0.276076	+	0.961136i	$0.589034\pi$
$264$	−1.65539	−0.101882
$265$	32.3034	1.98438
$266$	0	0
$267$	−8.99780	−0.550657
$268$	−10.7778	−0.658360
$269$	−4.82345	−0.294091	−0.147045	−	0.989130i	$-0.546976\pi$
$269$	−4.82345	−0.294091	−0.147045	+	0.989130i	$0.546976\pi$
$270$	10.7740	0.655688
$271$	7.42144	0.450820	0.225410	−	0.974264i	$-0.427628\pi$
$271$	7.42144	0.450820	0.225410	+	0.974264i	$0.427628\pi$
$272$	0.345312	0.0209376
$273$	0	0
$274$	−7.41938	−0.448221
$275$	6.14562	0.370595
$276$	−4.76558	−0.286854
$277$	3.81631	0.229300	0.114650	−	0.993406i	$-0.463425\pi$
$277$	3.81631	0.229300	0.114650	+	0.993406i	$0.463425\pi$
$278$	3.58536	0.215036
$279$	−23.4081	−1.40140
$280$	0	0
$281$	−8.54978	−0.510037	−0.255019	−	0.966936i	$-0.582082\pi$
$281$	−8.54978	−0.510037	−0.255019	+	0.966936i	$0.582082\pi$
$282$	−4.04714	−0.241004
$283$	15.2643	0.907371	0.453686	−	0.891162i	$-0.350109\pi$
$283$	15.2643	0.907371	0.453686	+	0.891162i	$0.350109\pi$
$284$	−4.39870	−0.261015
$285$	2.36215	0.139921
$286$	0	0
$287$	0	0
$288$	−14.6786	−0.864947
$289$	−13.2488	−0.779340
$290$	5.37770	0.315790
$291$	−0.334738	−0.0196227
$292$	13.9045	0.813702
$293$	5.93964	0.346997	0.173499	−	0.984834i	$-0.444493\pi$
$293$	5.93964	0.346997	0.173499	+	0.984834i	$0.444493\pi$
$294$	0	0
$295$	−1.75693	−0.102293
$296$	−29.5522	−1.71768
$297$	−3.30301	−0.191660
$298$	−4.99171	−0.289162
$299$	0	0
$300$	−5.77236	−0.333267
$301$	0	0
$302$	14.3472	0.825591
$303$	−3.95844	−0.227406
$304$	−0.185071	−0.0106146
$305$	−4.94262	−0.283013
$306$	−4.22467	−0.241508
$307$	−22.2133	−1.26778	−0.633891	−	0.773422i	$-0.718543\pi$
$307$	−22.2133	−1.26778	−0.633891	+	0.773422i	$0.718543\pi$
$308$	0	0
$309$	−2.73285	−0.155466
$310$	−26.7542	−1.51954
$311$	9.84259	0.558122	0.279061	−	0.960273i	$-0.409977\pi$
$311$	9.84259	0.558122	0.279061	+	0.960273i	$0.409977\pi$
$312$	0	0
$313$	−20.9125	−1.18205	−0.591023	−	0.806655i	$-0.701276\pi$
$313$	−20.9125	−1.18205	−0.591023	+	0.806655i	$0.701276\pi$
$314$	−1.65033	−0.0931337
$315$	0	0
$316$	15.4241	0.867673
$317$	25.3603	1.42438	0.712188	−	0.701989i	$-0.247705\pi$
$317$	25.3603	1.42438	0.712188	+	0.701989i	$0.247705\pi$
$318$	−5.28256	−0.296231
$319$	−1.64865	−0.0923065
$320$	−15.5498	−0.869259
$321$	−9.33870	−0.521236
$322$	0	0
$323$	−2.01048	−0.111866
$324$	−6.70475	−0.372486
$325$	0	0
$326$	10.1249	0.560768
$327$	2.77871	0.153663
$328$	−14.8637	−0.820709
$329$	0	0
$330$	−1.73923	−0.0957413
$331$	−1.78283	−0.0979935	−0.0489967	−	0.998799i	$-0.515602\pi$
$331$	−1.78283	−0.0979935	−0.0489967	+	0.998799i	$0.515602\pi$
$332$	1.93467	0.106179
$333$	−27.1656	−1.48866
$334$	−14.1083	−0.771971
$335$	−29.0781	−1.58871
$336$	0	0
$337$	9.56149	0.520848	0.260424	−	0.965494i	$-0.416138\pi$
$337$	9.56149	0.520848	0.260424	+	0.965494i	$0.416138\pi$
$338$	0	0
$339$	9.10468	0.494498
$340$	8.50240	0.461107
$341$	8.20207	0.444167
$342$	2.26422	0.122435
$343$	0	0
$344$	10.9138	0.588431
$345$	−12.8573	−0.692216
$346$	−8.49982	−0.456953
$347$	0.633389	0.0340021	0.0170010	−	0.999855i	$-0.494588\pi$
$347$	0.633389	0.0340021	0.0170010	+	0.999855i	$0.494588\pi$
$348$	1.54852	0.0830092
$349$	−30.5989	−1.63792	−0.818960	−	0.573850i	$-0.805449\pi$
$349$	−30.5989	−1.63792	−0.818960	+	0.573850i	$0.805449\pi$
$350$	0	0
$351$	0	0
$352$	5.14332	0.274140
$353$	1.10035	0.0585655	0.0292828	−	0.999571i	$-0.490678\pi$
$353$	1.10035	0.0585655	0.0292828	+	0.999571i	$0.490678\pi$
$354$	0.287311	0.0152704
$355$	−11.8675	−0.629862
$356$	17.3579	0.919969
$357$	0	0
$358$	7.81042	0.412793
$359$	9.77386	0.515845	0.257922	−	0.966166i	$-0.416962\pi$
$359$	9.77386	0.515845	0.257922	+	0.966166i	$0.416962\pi$
$360$	−24.5889	−1.29595
$361$	−17.9225	−0.943288
$362$	5.11091	0.268624
$363$	−6.74026	−0.353772
$364$	0	0
$365$	37.5139	1.96357
$366$	0.808264	0.0422487
$367$	−11.1473	−0.581882	−0.290941	−	0.956741i	$-0.593968\pi$
$367$	−11.1473	−0.581882	−0.290941	+	0.956741i	$0.593968\pi$
$368$	1.00736	0.0525121
$369$	−13.6633	−0.711283
$370$	−31.0488	−1.61415
$371$	0	0
$372$	−7.70391	−0.399429
$373$	−30.7301	−1.59115	−0.795573	−	0.605858i	$-0.792830\pi$
$373$	−30.7301	−1.59115	−0.795573	+	0.605858i	$0.792830\pi$
$374$	1.48030	0.0765445
$375$	−4.19567	−0.216663
$376$	20.0489	1.03394
$377$	0	0
$378$	0	0
$379$	−22.6572	−1.16382	−0.581912	−	0.813252i	$-0.697695\pi$
$379$	−22.6572	−1.16382	−0.581912	+	0.813252i	$0.697695\pi$
$380$	−4.55689	−0.233763
$381$	1.28606	0.0658866
$382$	1.12019	0.0573137
$383$	0.589263	0.0301099	0.0150550	−	0.999887i	$-0.495208\pi$
$383$	0.589263	0.0301099	0.0150550	+	0.999887i	$0.495208\pi$
$384$	−5.03161	−0.256769
$385$	0	0
$386$	13.9801	0.711567
$387$	10.0324	0.509975
$388$	0.645753	0.0327832
$389$	5.69945	0.288974	0.144487	−	0.989507i	$-0.453847\pi$
$389$	5.69945	0.288974	0.144487	+	0.989507i	$0.453847\pi$
$390$	0	0
$391$	10.9432	0.553422
$392$	0	0
$393$	−7.95809	−0.401433
$394$	21.7782	1.09717
$395$	41.6136	2.09381
$396$	2.93556	0.147518
$397$	25.5283	1.28123	0.640614	−	0.767863i	$-0.278680\pi$
$397$	25.5283	1.28123	0.640614	+	0.767863i	$0.278680\pi$
$398$	−21.6005	−1.08274
$399$	0	0
$400$	1.22017	0.0610086
$401$	−25.5011	−1.27347	−0.636733	−	0.771085i	$-0.719714\pi$
$401$	−25.5011	−1.27347	−0.636733	+	0.771085i	$0.719714\pi$
$402$	4.75514	0.237165
$403$	0	0
$404$	7.63635	0.379922
$405$	−18.0891	−0.898857
$406$	0	0
$407$	9.51867	0.471823
$408$	−3.57040	−0.176761
$409$	−0.146988	−0.00726807	−0.00363403	−	0.999993i	$-0.501157\pi$
$409$	−0.146988	−0.00726807	−0.00363403	+	0.999993i	$0.501157\pi$
$410$	−15.6164	−0.771241
$411$	−5.76398	−0.284316
$412$	5.27202	0.259734
$413$	0	0
$414$	−12.3243	−0.605709
$415$	5.21966	0.256223
$416$	0	0
$417$	2.78540	0.136402
$418$	−0.793372	−0.0388051
$419$	13.6959	0.669088	0.334544	−	0.942380i	$-0.391418\pi$
$419$	13.6959	0.669088	0.334544	+	0.942380i	$0.391418\pi$
$420$	0	0
$421$	−3.44169	−0.167738	−0.0838688	−	0.996477i	$-0.526728\pi$
$421$	−3.44169	−0.167738	−0.0838688	+	0.996477i	$0.526728\pi$
$422$	−4.84842	−0.236017
$423$	18.4297	0.896084
$424$	26.1689	1.27087
$425$	13.2551	0.642966
$426$	1.94069	0.0940267
$427$	0	0
$428$	18.0156	0.870816
$429$	0	0
$430$	11.4665	0.552964
$431$	−22.2910	−1.07372	−0.536861	−	0.843671i	$-0.680390\pi$
$431$	−22.2910	−1.07372	−0.536861	+	0.843671i	$0.680390\pi$
$432$	−0.655791	−0.0315518
$433$	−25.8963	−1.24449	−0.622247	−	0.782821i	$-0.713780\pi$
$433$	−25.8963	−1.24449	−0.622247	+	0.782821i	$0.713780\pi$
$434$	0	0
$435$	4.17783	0.200312
$436$	−5.36049	−0.256721
$437$	−5.86505	−0.280564
$438$	−6.13464	−0.293124
$439$	−27.9838	−1.33560	−0.667798	−	0.744343i	$-0.732763\pi$
$439$	−27.9838	−1.33560	−0.667798	+	0.744343i	$0.732763\pi$
$440$	8.61583	0.410744
$441$	0	0
$442$	0	0
$443$	33.2089	1.57780	0.788900	−	0.614521i	$-0.210651\pi$
$443$	33.2089	1.57780	0.788900	+	0.614521i	$0.210651\pi$
$444$	−8.94055	−0.424300
$445$	46.8310	2.22000
$446$	−2.00706	−0.0950370
$447$	−3.87796	−0.183421
$448$	0	0
$449$	−19.6864	−0.929059	−0.464529	−	0.885558i	$-0.653777\pi$
$449$	−19.6864	−0.929059	−0.464529	+	0.885558i	$0.653777\pi$
$450$	−14.9280	−0.703712
$451$	4.78755	0.225437
$452$	−17.5641	−0.826146
$453$	11.1461	0.523690
$454$	−22.3783	−1.05027
$455$	0	0
$456$	1.91357	0.0896111
$457$	0.746942	0.0349405	0.0174702	−	0.999847i	$-0.494439\pi$
$457$	0.746942	0.0349405	0.0174702	+	0.999847i	$0.494439\pi$
$458$	−0.0583493	−0.00272648
$459$	−7.12405	−0.332522
$460$	24.8035	1.15647
$461$	33.1710	1.54493	0.772464	−	0.635058i	$-0.219024\pi$
$461$	33.1710	1.54493	0.772464	+	0.635058i	$0.219024\pi$
$462$	0	0
$463$	30.7521	1.42917	0.714586	−	0.699548i	$-0.246615\pi$
$463$	30.7521	1.42917	0.714586	+	0.699548i	$0.246615\pi$
$464$	−0.327328	−0.0151958
$465$	−20.7848	−0.963874
$466$	12.4863	0.578414
$467$	−29.6065	−1.37003	−0.685013	−	0.728531i	$-0.740203\pi$
$467$	−29.6065	−1.37003	−0.685013	+	0.728531i	$0.740203\pi$
$468$	0	0
$469$	0	0
$470$	21.0642	0.971621
$471$	−1.28211	−0.0590766
$472$	−1.42329	−0.0655122
$473$	−3.51530	−0.161634
$474$	−6.80506	−0.312567
$475$	−7.10411	−0.325959
$476$	0	0
$477$	24.0555	1.10143
$478$	−2.85496	−0.130583
$479$	−14.0905	−0.643813	−0.321907	−	0.946771i	$-0.604324\pi$
$479$	−14.0905	−0.643813	−0.321907	+	0.946771i	$0.604324\pi$
$480$	−13.0337	−0.594903
$481$	0	0
$482$	7.30247	0.332618
$483$	0	0
$484$	13.0028	0.591038
$485$	1.74222	0.0791100
$486$	12.3500	0.560210
$487$	16.7955	0.761075	0.380537	−	0.924766i	$-0.375739\pi$
$487$	16.7955	0.761075	0.380537	+	0.924766i	$0.375739\pi$
$488$	−4.00400	−0.181253
$489$	7.86587	0.355707
$490$	0	0
$491$	21.6690	0.977908	0.488954	−	0.872310i	$-0.337379\pi$
$491$	21.6690	0.977908	0.488954	+	0.872310i	$0.337379\pi$
$492$	−4.49677	−0.202730
$493$	−3.55586	−0.160148
$494$	0	0
$495$	7.92003	0.355979
$496$	1.62847	0.0731203
$497$	0	0
$498$	−0.853570	−0.0382494
$499$	23.3048	1.04327	0.521633	−	0.853170i	$-0.325323\pi$
$499$	23.3048	1.04327	0.521633	+	0.853170i	$0.325323\pi$
$500$	8.09399	0.361974
$501$	−10.9605	−0.489677
$502$	18.3120	0.817306
$503$	−43.8829	−1.95664	−0.978322	−	0.207090i	$-0.933601\pi$
$503$	−43.8829	−1.95664	−0.978322	+	0.207090i	$0.933601\pi$
$504$	0	0
$505$	20.6026	0.916802
$506$	4.31839	0.191976
$507$	0	0
$508$	−2.48097	−0.110075
$509$	−19.9242	−0.883125	−0.441563	−	0.897230i	$-0.645576\pi$
$509$	−19.9242	−0.883125	−0.441563	+	0.897230i	$0.645576\pi$
$510$	−3.75123	−0.166107
$511$	0	0
$512$	2.01529	0.0890641
$513$	3.81816	0.168576
$514$	4.20279	0.185377
$515$	14.2237	0.626771
$516$	3.30179	0.145353
$517$	−6.45768	−0.284009
$518$	0	0
$519$	−6.60335	−0.289855
$520$	0	0
$521$	−16.5241	−0.723933	−0.361967	−	0.932191i	$-0.617895\pi$
$521$	−16.5241	−0.723933	−0.361967	+	0.932191i	$0.617895\pi$
$522$	4.00464	0.175278
$523$	−11.9962	−0.524556	−0.262278	−	0.964992i	$-0.584474\pi$
$523$	−11.9962	−0.524556	−0.262278	+	0.964992i	$0.584474\pi$
$524$	15.3522	0.670664
$525$	0	0
$526$	−7.62131	−0.332305
$527$	17.6905	0.770611
$528$	0.105863	0.00460709
$529$	8.92395	0.387998
$530$	27.4942	1.19427
$531$	−1.30835	−0.0567774
$532$	0	0
$533$	0	0
$534$	−7.65826	−0.331405
$535$	48.6053	2.10139
$536$	−23.5561	−1.01747
$537$	6.06776	0.261843
$538$	−4.10536	−0.176995
$539$	0	0
$540$	−16.1471	−0.694861
$541$	−36.2317	−1.55772	−0.778860	−	0.627197i	$-0.784202\pi$
$541$	−36.2317	−1.55772	−0.778860	+	0.627197i	$0.784202\pi$
$542$	6.31658	0.271320
$543$	3.97057	0.170394
$544$	11.0933	0.475621
$545$	−14.4624	−0.619500
$546$	0	0
$547$	−7.34857	−0.314202	−0.157101	−	0.987583i	$-0.550215\pi$
$547$	−7.34857	−0.314202	−0.157101	+	0.987583i	$0.550215\pi$
$548$	11.1195	0.475000
$549$	−3.68064	−0.157086
$550$	5.23069	0.223037
$551$	1.90578	0.0811888
$552$	−10.4157	−0.443322
$553$	0	0
$554$	3.24816	0.138001
$555$	−24.1213	−1.02389
$556$	−5.37340	−0.227883
$557$	−10.8280	−0.458796	−0.229398	−	0.973333i	$-0.573676\pi$
$557$	−10.8280	−0.458796	−0.229398	+	0.973333i	$0.573676\pi$
$558$	−19.9232	−0.843417
$559$	0	0
$560$	0	0
$561$	1.15002	0.0485538
$562$	−7.27694	−0.306959
$563$	−13.8599	−0.584127	−0.292064	−	0.956399i	$-0.594342\pi$
$563$	−13.8599	−0.584127	−0.292064	+	0.956399i	$0.594342\pi$
$564$	6.06547	0.255402
$565$	−47.3873	−1.99360
$566$	12.9919	0.546089
$567$	0	0
$568$	−9.61384	−0.403388
$569$	27.4120	1.14917	0.574586	−	0.818444i	$-0.305163\pi$
$569$	27.4120	1.14917	0.574586	+	0.818444i	$0.305163\pi$
$570$	2.01048	0.0842099
$571$	−0.207758	−0.00869439	−0.00434719	−	0.999991i	$-0.501384\pi$
$571$	−0.207758	−0.00869439	−0.00434719	+	0.999991i	$0.501384\pi$
$572$	0	0
$573$	0.870251	0.0363552
$574$	0	0
$575$	38.6682	1.61258
$576$	−11.5795	−0.482480
$577$	3.32656	0.138487	0.0692434	−	0.997600i	$-0.477942\pi$
$577$	3.32656	0.138487	0.0692434	+	0.997600i	$0.477942\pi$
$578$	−11.2764	−0.469035
$579$	10.8609	0.451362
$580$	−8.05959	−0.334656
$581$	0	0
$582$	−0.284904	−0.0118097
$583$	−8.42893	−0.349091
$584$	30.3899	1.25754
$585$	0	0
$586$	5.05538	0.208836
$587$	15.0810	0.622461	0.311230	−	0.950335i	$-0.399259\pi$
$587$	15.0810	0.622461	0.311230	+	0.950335i	$0.399259\pi$
$588$	0	0
$589$	−9.48129	−0.390670
$590$	−1.49537	−0.0615635
$591$	16.9191	0.695957
$592$	1.88987	0.0776732
$593$	−25.8491	−1.06149	−0.530747	−	0.847530i	$-0.678089\pi$
$593$	−25.8491	−1.06149	−0.530747	+	0.847530i	$0.678089\pi$
$594$	−2.81128	−0.115348
$595$	0	0
$596$	7.48110	0.306438
$597$	−16.7810	−0.686803
$598$	0	0
$599$	−35.5469	−1.45241	−0.726203	−	0.687480i	$-0.758717\pi$
$599$	−35.5469	−1.45241	−0.726203	+	0.687480i	$0.758717\pi$
$600$	−12.6161	−0.515052
$601$	−27.2947	−1.11338	−0.556688	−	0.830722i	$-0.687928\pi$
$601$	−27.2947	−1.11338	−0.556688	+	0.830722i	$0.687928\pi$
$602$	0	0
$603$	−21.6538	−0.881810
$604$	−21.5023	−0.874915
$605$	35.0811	1.42625
$606$	−3.36913	−0.136862
$607$	−38.9258	−1.57995	−0.789976	−	0.613138i	$-0.789907\pi$
$607$	−38.9258	−1.57995	−0.789976	+	0.613138i	$0.789907\pi$
$608$	−5.94549	−0.241121
$609$	0	0
$610$	−4.20679	−0.170328
$611$	0	0
$612$	6.33152	0.255937
$613$	−0.886645	−0.0358113	−0.0179056	−	0.999840i	$-0.505700\pi$
$613$	−0.886645	−0.0358113	−0.0179056	+	0.999840i	$0.505700\pi$
$614$	−18.9063	−0.762997
$615$	−12.1321	−0.489214
$616$	0	0
$617$	−34.7888	−1.40054	−0.700272	−	0.713876i	$-0.746938\pi$
$617$	−34.7888	−1.40054	−0.700272	+	0.713876i	$0.746938\pi$
$618$	−2.32600	−0.0935653
$619$	−2.05562	−0.0826221	−0.0413111	−	0.999146i	$-0.513153\pi$
$619$	−2.05562	−0.0826221	−0.0413111	+	0.999146i	$0.513153\pi$
$620$	40.0967	1.61032
$621$	−20.7825	−0.833975
$622$	8.37728	0.335898
$623$	0	0
$624$	0	0
$625$	−12.3816	−0.495265
$626$	−17.7992	−0.711398
$627$	−0.616356	−0.0246149
$628$	2.47336	0.0986979
$629$	20.5302	0.818593
$630$	0	0
$631$	45.2337	1.80073	0.900363	−	0.435139i	$-0.143301\pi$
$631$	45.2337	1.80073	0.900363	+	0.435139i	$0.143301\pi$
$632$	33.7111	1.34095
$633$	−3.76665	−0.149711
$634$	21.5848	0.857241
$635$	−6.69355	−0.265625
$636$	7.91700	0.313929
$637$	0	0
$638$	−1.40321	−0.0555534
$639$	−8.83744	−0.349604
$640$	26.1881	1.03518
$641$	−19.0619	−0.752902	−0.376451	−	0.926437i	$-0.622856\pi$
$641$	−19.0619	−0.752902	−0.376451	+	0.926437i	$0.622856\pi$
$642$	−7.94841	−0.313699
$643$	10.5351	0.415464	0.207732	−	0.978186i	$-0.433392\pi$
$643$	10.5351	0.415464	0.207732	+	0.978186i	$0.433392\pi$
$644$	0	0
$645$	8.90811	0.350756
$646$	−1.71117	−0.0673252
$647$	−24.1608	−0.949860	−0.474930	−	0.880024i	$-0.657527\pi$
$647$	−24.1608	−0.949860	−0.474930	+	0.880024i	$0.657527\pi$
$648$	−14.6540	−0.575663
$649$	0.458438	0.0179952
$650$	0	0
$651$	0	0
$652$	−15.1743	−0.594271
$653$	−33.6890	−1.31835	−0.659176	−	0.751988i	$-0.729095\pi$
$653$	−33.6890	−1.31835	−0.659176	+	0.751988i	$0.729095\pi$
$654$	2.36503	0.0924799
$655$	41.4196	1.61840
$656$	0.950537	0.0371122
$657$	27.9357	1.08987
$658$	0	0
$659$	−4.20059	−0.163632	−0.0818159	−	0.996647i	$-0.526072\pi$
$659$	−4.20059	−0.163632	−0.0818159	+	0.996647i	$0.526072\pi$
$660$	2.60659	0.101461
$661$	−17.6726	−0.687385	−0.343693	−	0.939082i	$-0.611678\pi$
$661$	−17.6726	−0.687385	−0.343693	+	0.939082i	$0.611678\pi$
$662$	−1.51742	−0.0589760
$663$	0	0
$664$	4.22844	0.164095
$665$	0	0
$666$	−23.1213	−0.895932
$667$	−10.3733	−0.401655
$668$	21.1441	0.818091
$669$	−1.55925	−0.0602839
$670$	−24.7491	−0.956143
$671$	1.28968	0.0497875
$672$	0	0
$673$	−20.6103	−0.794469	−0.397235	−	0.917717i	$-0.630030\pi$
$673$	−20.6103	−0.794469	−0.397235	+	0.917717i	$0.630030\pi$
$674$	8.13803	0.313465
$675$	−25.1731	−0.968911
$676$	0	0
$677$	−21.3074	−0.818909	−0.409455	−	0.912330i	$-0.634281\pi$
$677$	−21.3074	−0.818909	−0.409455	+	0.912330i	$0.634281\pi$
$678$	7.74922	0.297607
$679$	0	0
$680$	18.5829	0.712623
$681$	−17.3853	−0.666206
$682$	6.98099	0.267316
$683$	6.69757	0.256275	0.128138	−	0.991756i	$-0.459100\pi$
$683$	6.69757	0.256275	0.128138	+	0.991756i	$0.459100\pi$
$684$	−3.39340	−0.129750
$685$	29.9999	1.14624
$686$	0	0
$687$	−0.0453305	−0.00172946
$688$	−0.697939	−0.0266087
$689$	0	0
$690$	−10.9432	−0.416601
$691$	24.9263	0.948242	0.474121	−	0.880460i	$-0.342766\pi$
$691$	24.9263	0.948242	0.474121	+	0.880460i	$0.342766\pi$
$692$	12.7387	0.484253
$693$	0	0
$694$	0.539093	0.0204637
$695$	−14.4972	−0.549911
$696$	3.38445	0.128287
$697$	10.3260	0.391123
$698$	−26.0435	−0.985761
$699$	9.70033	0.366900
$700$	0	0
$701$	−4.94583	−0.186801	−0.0934007	−	0.995629i	$-0.529774\pi$
$701$	−4.94583	−0.186801	−0.0934007	+	0.995629i	$0.529774\pi$
$702$	0	0
$703$	−11.0032	−0.414995
$704$	4.05741	0.152919
$705$	16.3644	0.616319
$706$	0.936533	0.0352469
$707$	0	0
$708$	−0.430594	−0.0161827
$709$	4.64497	0.174446	0.0872228	−	0.996189i	$-0.472201\pi$
$709$	4.64497	0.174446	0.0872228	+	0.996189i	$0.472201\pi$
$710$	−10.1007	−0.379074
$711$	30.9886	1.16216
$712$	37.9377	1.42178
$713$	51.6074	1.93271
$714$	0	0
$715$	0	0
$716$	−11.7055	−0.437455
$717$	−2.21797	−0.0828315
$718$	8.31878	0.310454
$719$	31.7413	1.18375	0.591875	−	0.806030i	$-0.298388\pi$
$719$	31.7413	1.18375	0.591875	+	0.806030i	$0.298388\pi$
$720$	1.57247	0.0586025
$721$	0	0
$722$	−15.2543	−0.567705
$723$	5.67315	0.210987
$724$	−7.65975	−0.284672
$725$	−12.5647	−0.466643
$726$	−5.73681	−0.212913
$727$	47.8755	1.77560	0.887801	−	0.460227i	$-0.152232\pi$
$727$	47.8755	1.77560	0.887801	+	0.460227i	$0.152232\pi$
$728$	0	0
$729$	−6.17412	−0.228671
$730$	31.9290	1.18175
$731$	−7.58192	−0.280427
$732$	−1.21135	−0.0447728
$733$	7.60208	0.280789	0.140395	−	0.990096i	$-0.455163\pi$
$733$	7.60208	0.280789	0.140395	+	0.990096i	$0.455163\pi$
$734$	−9.48771	−0.350198
$735$	0	0
$736$	32.3618	1.19287
$737$	7.58737	0.279484
$738$	−11.6292	−0.428076
$739$	33.4236	1.22951	0.614754	−	0.788719i	$-0.289255\pi$
$739$	33.4236	1.22951	0.614754	+	0.788719i	$0.289255\pi$
$740$	46.5330	1.71059
$741$	0	0
$742$	0	0
$743$	2.93823	0.107793	0.0538966	−	0.998547i	$-0.482836\pi$
$743$	2.93823	0.107793	0.0538966	+	0.998547i	$0.482836\pi$
$744$	−16.8378	−0.617302
$745$	20.1837	0.739474
$746$	−26.1552	−0.957609
$747$	3.88695	0.142216
$748$	−2.21853	−0.0811176
$749$	0	0
$750$	−3.57104	−0.130396
$751$	1.19678	0.0436711	0.0218355	−	0.999762i	$-0.493049\pi$
$751$	1.19678	0.0436711	0.0218355	+	0.999762i	$0.493049\pi$
$752$	−1.28213	−0.0467545
$753$	14.2263	0.518434
$754$	0	0
$755$	−58.0123	−2.11128
$756$	0	0
$757$	11.5464	0.419662	0.209831	−	0.977738i	$-0.432709\pi$
$757$	11.5464	0.419662	0.209831	+	0.977738i	$0.432709\pi$
$758$	−19.2841	−0.700432
$759$	3.35487	0.121774
$760$	−9.95959	−0.361272
$761$	−34.6497	−1.25605	−0.628026	−	0.778192i	$-0.716137\pi$
$761$	−34.6497	−1.25605	−0.628026	+	0.778192i	$0.716137\pi$
$762$	1.09459	0.0396530
$763$	0	0
$764$	−1.67883	−0.0607378
$765$	17.0822	0.617608
$766$	0.501537	0.0181213
$767$	0	0
$768$	−10.2578	−0.370146
$769$	6.54874	0.236154	0.118077	−	0.993004i	$-0.462327\pi$
$769$	6.54874	0.236154	0.118077	+	0.993004i	$0.462327\pi$
$770$	0	0
$771$	3.26507	0.117589
$772$	−20.9520	−0.754079
$773$	−33.9275	−1.22029	−0.610143	−	0.792291i	$-0.708888\pi$
$773$	−33.9275	−1.22029	−0.610143	+	0.792291i	$0.708888\pi$
$774$	8.53882	0.306922
$775$	62.5100	2.24542
$776$	1.41137	0.0506651
$777$	0	0
$778$	4.85095	0.173915
$779$	−5.53423	−0.198284
$780$	0	0
$781$	3.09659	0.110805
$782$	9.31405	0.333070
$783$	6.75302	0.241333
$784$	0	0
$785$	6.67303	0.238171
$786$	−6.77333	−0.241597
$787$	−12.9743	−0.462485	−0.231243	−	0.972896i	$-0.574279\pi$
$787$	−12.9743	−0.462485	−0.231243	+	0.972896i	$0.574279\pi$
$788$	−32.6390	−1.16272
$789$	−5.92085	−0.210788
$790$	35.4184	1.26013
$791$	0	0
$792$	6.41600	0.227983
$793$	0	0
$794$	21.7278	0.771090
$795$	21.3597	0.757552
$796$	32.3728	1.14742
$797$	4.41913	0.156533	0.0782667	−	0.996932i	$-0.475061\pi$
$797$	4.41913	0.156533	0.0782667	+	0.996932i	$0.475061\pi$
$798$	0	0
$799$	−13.9282	−0.492743
$800$	39.1985	1.38588
$801$	34.8739	1.23221
$802$	−21.7047	−0.766418
$803$	−9.78852	−0.345429
$804$	−7.12655	−0.251334
$805$	0	0
$806$	0	0
$807$	−3.18937	−0.112271
$808$	16.6901	0.587155
$809$	11.4716	0.403320	0.201660	−	0.979456i	$-0.435366\pi$
$809$	11.4716	0.403320	0.201660	+	0.979456i	$0.435366\pi$
$810$	−15.3961	−0.540965
$811$	−23.8664	−0.838063	−0.419032	−	0.907972i	$-0.637630\pi$
$811$	−23.8664	−0.838063	−0.419032	+	0.907972i	$0.637630\pi$
$812$	0	0
$813$	4.90723	0.172104
$814$	8.10159	0.283960
$815$	−40.9396	−1.43405
$816$	0.228329	0.00799310
$817$	4.06355	0.142166
$818$	−0.125105	−0.00437419
$819$	0	0
$820$	23.4044	0.817318
$821$	−30.9694	−1.08084	−0.540420	−	0.841395i	$-0.681735\pi$
$821$	−30.9694	−1.08084	−0.540420	+	0.841395i	$0.681735\pi$
$822$	−4.90587	−0.171112
$823$	−8.61357	−0.300250	−0.150125	−	0.988667i	$-0.547968\pi$
$823$	−8.61357	−0.300250	−0.150125	+	0.988667i	$0.547968\pi$
$824$	11.5226	0.401408
$825$	4.06362	0.141477
$826$	0	0
$827$	−22.9128	−0.796756	−0.398378	−	0.917221i	$-0.630427\pi$
$827$	−22.9128	−0.796756	−0.398378	+	0.917221i	$0.630427\pi$
$828$	18.4706	0.641896
$829$	−42.5611	−1.47821	−0.739104	−	0.673591i	$-0.764751\pi$
$829$	−42.5611	−1.47821	−0.739104	+	0.673591i	$0.764751\pi$
$830$	4.44259	0.154204
$831$	2.52344	0.0875370
$832$	0	0
$833$	0	0
$834$	2.37073	0.0820915
$835$	57.0460	1.97416
$836$	1.18903	0.0411235
$837$	−33.5965	−1.16126
$838$	11.6569	0.402682
$839$	−1.84105	−0.0635601	−0.0317800	−	0.999495i	$-0.510118\pi$
$839$	−1.84105	−0.0635601	−0.0317800	+	0.999495i	$0.510118\pi$
$840$	0	0
$841$	−25.6293	−0.883770
$842$	−2.92931	−0.100951
$843$	−5.65332	−0.194711
$844$	7.26635	0.250118
$845$	0	0
$846$	15.6860	0.539296
$847$	0	0
$848$	−1.67351	−0.0574686
$849$	10.0931	0.346396
$850$	11.2817	0.386960
$851$	59.8915	2.05305
$852$	−2.90852	−0.0996442
$853$	27.0293	0.925466	0.462733	−	0.886498i	$-0.346869\pi$
$853$	27.0293	0.925466	0.462733	+	0.886498i	$0.346869\pi$
$854$	0	0
$855$	−9.15526	−0.313103
$856$	39.3750	1.34581
$857$	16.7854	0.573377	0.286688	−	0.958024i	$-0.407446\pi$
$857$	16.7854	0.573377	0.286688	+	0.958024i	$0.407446\pi$
$858$	0	0
$859$	51.6116	1.76096	0.880482	−	0.474079i	$-0.157219\pi$
$859$	51.6116	1.76096	0.880482	+	0.474079i	$0.157219\pi$
$860$	−17.1849	−0.586000
$861$	0	0
$862$	−18.9725	−0.646205
$863$	−21.9614	−0.747573	−0.373787	−	0.927515i	$-0.621941\pi$
$863$	−21.9614	−0.747573	−0.373787	+	0.927515i	$0.621941\pi$
$864$	−21.0676	−0.716733
$865$	34.3685	1.16857
$866$	−22.0410	−0.748983
$867$	−8.76040	−0.297519
$868$	0	0
$869$	−10.8583	−0.368341
$870$	3.55586	0.120555
$871$	0	0
$872$	−11.7159	−0.396752
$873$	1.29739	0.0439098
$874$	−4.99190	−0.168853
$875$	0	0
$876$	9.19401	0.310637
$877$	9.61745	0.324758	0.162379	−	0.986728i	$-0.448083\pi$
$877$	9.61745	0.324758	0.162379	+	0.986728i	$0.448083\pi$
$878$	−23.8178	−0.803811
$879$	3.92743	0.132469
$880$	−0.550986	−0.0185737
$881$	28.9726	0.976110	0.488055	−	0.872813i	$-0.337707\pi$
$881$	28.9726	0.976110	0.488055	+	0.872813i	$0.337707\pi$
$882$	0	0
$883$	6.60727	0.222352	0.111176	−	0.993801i	$-0.464538\pi$
$883$	6.60727	0.222352	0.111176	+	0.993801i	$0.464538\pi$
$884$	0	0
$885$	−1.16173	−0.0390510
$886$	28.2649	0.949578
$887$	−31.4144	−1.05479	−0.527397	−	0.849619i	$-0.676832\pi$
$887$	−31.4144	−1.05479	−0.527397	+	0.849619i	$0.676832\pi$
$888$	−19.5406	−0.655739
$889$	0	0
$890$	39.8591	1.33608
$891$	4.72001	0.158126
$892$	3.00799	0.100715
$893$	7.46484	0.249801
$894$	−3.30063	−0.110390
$895$	−31.5810	−1.05564
$896$	0	0
$897$	0	0
$898$	−16.7556	−0.559142
$899$	−16.7692	−0.559283
$900$	22.3726	0.745755
$901$	−18.1798	−0.605658
$902$	4.07480	0.135676
$903$	0	0
$904$	−38.3883	−1.27678
$905$	−20.6657	−0.686951
$906$	9.48673	0.315176
$907$	9.73642	0.323292	0.161646	−	0.986849i	$-0.448320\pi$
$907$	9.73642	0.323292	0.161646	+	0.986849i	$0.448320\pi$
$908$	33.5385	1.11301
$909$	15.3422	0.508869
$910$	0	0
$911$	−38.4372	−1.27348	−0.636740	−	0.771078i	$-0.719718\pi$
$911$	−38.4372	−1.27348	−0.636740	+	0.771078i	$0.719718\pi$
$912$	−0.122373	−0.00405219
$913$	−1.36197	−0.0450746
$914$	0.635741	0.0210285
$915$	−3.26817	−0.108042
$916$	0.0874483	0.00288937
$917$	0	0
$918$	−6.06346	−0.200124
$919$	54.2804	1.79055	0.895273	−	0.445519i	$-0.146981\pi$
$919$	54.2804	1.79055	0.895273	+	0.445519i	$0.146981\pi$
$920$	54.2108	1.78728
$921$	−14.6880	−0.483985
$922$	28.2327	0.929795
$923$	0	0
$924$	0	0
$925$	72.5442	2.38524
$926$	26.1739	0.860128
$927$	10.5920	0.347888
$928$	−10.5155	−0.345190
$929$	38.1920	1.25304	0.626519	−	0.779406i	$-0.284479\pi$
$929$	38.1920	1.25304	0.626519	+	0.779406i	$0.284479\pi$
$930$	−17.6905	−0.580095
$931$	0	0
$932$	−18.7132	−0.612971
$933$	6.50815	0.213067
$934$	−25.1988	−0.824531
$935$	−5.98552	−0.195747
$936$	0	0
$937$	−19.0376	−0.621931	−0.310966	−	0.950421i	$-0.600652\pi$
$937$	−19.0376	−0.621931	−0.310966	+	0.950421i	$0.600652\pi$
$938$	0	0
$939$	−13.8278	−0.451255
$940$	−31.5691	−1.02967
$941$	−46.1622	−1.50484	−0.752422	−	0.658682i	$-0.771114\pi$
$941$	−46.1622	−1.50484	−0.752422	+	0.658682i	$0.771114\pi$
$942$	−1.09124	−0.0355545
$943$	30.1232	0.980948
$944$	0.0910198	0.00296244
$945$	0	0
$946$	−2.99196	−0.0972770
$947$	9.19374	0.298756	0.149378	−	0.988780i	$-0.452273\pi$
$947$	9.19374	0.298756	0.149378	+	0.988780i	$0.452273\pi$
$948$	10.1988	0.331241
$949$	0	0
$950$	−6.04648	−0.196174
$951$	16.7688	0.543766
$952$	0	0
$953$	−44.6463	−1.44624	−0.723118	−	0.690725i	$-0.757292\pi$
$953$	−44.6463	−1.44624	−0.723118	+	0.690725i	$0.757292\pi$
$954$	20.4743	0.662879
$955$	−4.52941	−0.146568
$956$	4.27874	0.138385
$957$	−1.09012	−0.0352387
$958$	−11.9928	−0.387470
$959$	0	0
$960$	−10.2819	−0.331846
$961$	52.4271	1.69120
$962$	0	0
$963$	36.1952	1.16637
$964$	−10.9442	−0.352490
$965$	−56.5276	−1.81969
$966$	0	0
$967$	−13.8268	−0.444639	−0.222320	−	0.974974i	$-0.571363\pi$
$967$	−13.8268	−0.444639	−0.222320	+	0.974974i	$0.571363\pi$
$968$	28.4191	0.913426
$969$	−1.32938	−0.0427058
$970$	1.48284	0.0476113
$971$	7.26873	0.233265	0.116632	−	0.993175i	$-0.462790\pi$
$971$	7.26873	0.233265	0.116632	+	0.993175i	$0.462790\pi$
$972$	−18.5091	−0.593679
$973$	0	0
$974$	14.2950	0.458043
$975$	0	0
$976$	0.256057	0.00819619
$977$	−42.8101	−1.36962	−0.684808	−	0.728723i	$-0.740114\pi$
$977$	−42.8101	−1.36962	−0.684808	+	0.728723i	$0.740114\pi$
$978$	6.69484	0.214077
$979$	−12.2196	−0.390541
$980$	0	0
$981$	−10.7698	−0.343852
$982$	18.4430	0.588541
$983$	46.3088	1.47702	0.738511	−	0.674242i	$-0.235529\pi$
$983$	46.3088	1.47702	0.738511	+	0.674242i	$0.235529\pi$
$984$	−9.82820	−0.313312
$985$	−88.0588	−2.80579
$986$	−3.02648	−0.0963829
$987$	0	0
$988$	0	0
$989$	−22.1182	−0.703319
$990$	6.74094	0.214241
$991$	−58.2324	−1.84981	−0.924907	−	0.380194i	$-0.875857\pi$
$991$	−58.2324	−1.84981	−0.924907	+	0.380194i	$0.875857\pi$
$992$	52.3151	1.66101
$993$	−1.17885	−0.0374097
$994$	0	0
$995$	87.3406	2.76888
$996$	1.27925	0.0405346
$997$	−4.49479	−0.142351	−0.0711757	−	0.997464i	$-0.522675\pi$
$997$	−4.49479	−0.142351	−0.0711757	+	0.997464i	$0.522675\pi$
$998$	19.8353	0.627875
$999$	−38.9894	−1.23357

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.ce.1.4		6
7.2	even	3		1183.2.e.g.508.3		12
7.4	even	3		1183.2.e.g.170.3		12
7.6	odd	2		8281.2.a.cf.1.4		6
13.4	even	6		637.2.f.k.393.4		12
13.10	even	6		637.2.f.k.295.4		12
13.12	even	2		8281.2.a.bz.1.3		6
91.4	even	6		91.2.h.b.16.3	yes	12
91.10	odd	6		637.2.g.l.373.4		12
91.17	odd	6		637.2.h.l.471.3		12
91.23	even	6		91.2.h.b.74.3	yes	12
91.25	even	6		1183.2.e.h.170.4		12
91.30	even	6		91.2.g.b.81.4	yes	12
91.51	even	6		1183.2.e.h.508.4		12
91.62	odd	6		637.2.f.j.295.4		12
91.69	odd	6		637.2.f.j.393.4		12
91.75	odd	6		637.2.h.l.165.3		12
91.82	odd	6		637.2.g.l.263.4		12
91.88	even	6		91.2.g.b.9.4	✓	12
91.90	odd	2		8281.2.a.ca.1.3		6
273.23	odd	6		819.2.s.d.802.4		12
273.95	odd	6		819.2.s.d.289.4		12
273.179	odd	6		819.2.n.d.100.3		12
273.212	odd	6		819.2.n.d.172.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.g.b.9.4	✓	12	91.88	even	6
91.2.g.b.81.4	yes	12	91.30	even	6
91.2.h.b.16.3	yes	12	91.4	even	6
91.2.h.b.74.3	yes	12	91.23	even	6
637.2.f.j.295.4		12	91.62	odd	6
637.2.f.j.393.4		12	91.69	odd	6
637.2.f.k.295.4		12	13.10	even	6
637.2.f.k.393.4		12	13.4	even	6
637.2.g.l.263.4		12	91.82	odd	6
637.2.g.l.373.4		12	91.10	odd	6
637.2.h.l.165.3		12	91.75	odd	6
637.2.h.l.471.3		12	91.17	odd	6
819.2.n.d.100.3		12	273.179	odd	6
819.2.n.d.172.3		12	273.212	odd	6
819.2.s.d.289.4		12	273.95	odd	6
819.2.s.d.802.4		12	273.23	odd	6
1183.2.e.g.170.3		12	7.4	even	3
1183.2.e.g.508.3		12	7.2	even	3
1183.2.e.h.170.4		12	91.25	even	6
1183.2.e.h.508.4		12	91.51	even	6
8281.2.a.bz.1.3		6	13.12	even	2
8281.2.a.ca.1.3		6	91.90	odd	2
8281.2.a.ce.1.4		6	1.1	even	1	trivial
8281.2.a.cf.1.4		6	7.6	odd	2

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

\(f(q)\)	\(=\)	\(q+0.851125 q^{2} +0.661223 q^{3} -1.27559 q^{4} -3.44148 q^{5} +0.562784 q^{6} -2.78793 q^{8} -2.56278 q^{9} -2.92913 q^{10} +0.897986 q^{11} -0.843447 q^{12} -2.27559 q^{15} +0.178289 q^{16} +1.93681 q^{17} -2.18125 q^{18} -1.03804 q^{19} +4.38990 q^{20} +0.764299 q^{22} +5.65013 q^{23} -1.84345 q^{24} +6.84378 q^{25} -3.67824 q^{27} -1.83594 q^{29} -1.93681 q^{30} +9.13385 q^{31} +5.72761 q^{32} +0.593769 q^{33} +1.64847 q^{34} +3.26905 q^{36} +10.6000 q^{37} -0.883501 q^{38} +9.59462 q^{40} +5.33143 q^{41} -3.91465 q^{43} -1.14546 q^{44} +8.81977 q^{45} +4.80897 q^{46} -7.19129 q^{47} +0.117889 q^{48} +5.82491 q^{50} +1.28066 q^{51} -9.38648 q^{53} -3.13065 q^{54} -3.09040 q^{55} -0.686375 q^{57} -1.56261 q^{58} +0.510517 q^{59} +2.90270 q^{60} +1.43619 q^{61} +7.77405 q^{62} +4.51834 q^{64} +0.505372 q^{66} +8.44932 q^{67} -2.47057 q^{68} +3.73600 q^{69} +3.44837 q^{71} +7.14487 q^{72} -10.9005 q^{73} +9.02195 q^{74} +4.52526 q^{75} +1.32411 q^{76} -12.0918 q^{79} -0.613579 q^{80} +5.25621 q^{81} +4.53771 q^{82} -1.51669 q^{83} -6.66549 q^{85} -3.33185 q^{86} -1.21396 q^{87} -2.50353 q^{88} -13.6078 q^{89} +7.50673 q^{90} -7.20722 q^{92} +6.03951 q^{93} -6.12069 q^{94} +3.57239 q^{95} +3.78723 q^{96} -0.506241 q^{97} -2.30134 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} - q^{3} + 4 q^{4} + q^{5} - 9 q^{6} + 3 q^{8} - 3 q^{9} - 4 q^{10} + 4 q^{11} - 5 q^{12} - 2 q^{15} - 8 q^{16} - 5 q^{17} + 3 q^{18} - q^{19} - q^{20} + 5 q^{22} + q^{23} - 11 q^{24}+ \cdots + 10 q^{99}+O(q^{100}) \) 6 * q + 2 * q^2 - q^3 + 4 * q^4 + q^5 - 9 * q^6 + 3 * q^8 - 3 * q^9 - 4 * q^10 + 4 * q^11 - 5 * q^12 - 2 * q^15 - 8 * q^16 - 5 * q^17 + 3 * q^18 - q^19 - q^20 + 5 * q^22 + q^23 - 11 * q^24 - 7 * q^25 - 4 * q^27 - 3 * q^29 + 5 * q^30 + 16 * q^31 + 8 * q^32 + 16 * q^33 - 16 * q^34 + 21 * q^36 - 13 * q^37 + 17 * q^38 + 5 * q^40 - 8 * q^41 + 11 * q^43 + 21 * q^44 - 7 * q^45 + 16 * q^46 - q^47 - 21 * q^48 + 6 * q^50 + 20 * q^51 + 2 * q^53 - 18 * q^54 - 9 * q^55 - 21 * q^57 - 8 * q^58 + 13 * q^59 + 20 * q^60 + 5 * q^61 - 5 * q^62 - 15 * q^64 - 18 * q^66 - 11 * q^67 - 29 * q^68 - 23 * q^69 + 6 * q^71 + 25 * q^72 - 30 * q^73 + 3 * q^74 + 3 * q^75 - 9 * q^76 - 7 * q^79 - 7 * q^80 + 6 * q^81 - q^82 + 27 * q^83 - q^85 - 7 * q^86 - 16 * q^87 + 4 * q^89 - 8 * q^90 + 27 * q^92 - 7 * q^93 - 45 * q^94 + 6 * q^95 + 19 * q^96 - 35 * q^97 + 10 * q^99

Introduction

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