LMFDB - Embedded newform 539.2.a.l.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [539,2,Mod(1,539)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, 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("539.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	539.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:	$4.30393666895$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 26x^{8} + 245x^{6} - 1038x^{4} + 1884x^{2} - 968 $ x^10 - 26x^8 + 245x^6 - 1038x^4 + 1884x^2 - 968
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-3.27614$ of defining polynomial
Character	$\chi$	$=$	539.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.70296 q^{2} -3.27614 q^{3} +0.900071 q^{4} +0.246676 q^{5} -5.57913 q^{6} -1.87313 q^{8} +7.73309 q^{9} +O(q^{10})$	\(q+1.70296 q^{2} -3.27614 q^{3} +0.900071 q^{4} +0.246676 q^{5} -5.57913 q^{6} -1.87313 q^{8} +7.73309 q^{9} +0.420079 q^{10} +1.00000 q^{11} -2.94876 q^{12} +3.17078 q^{13} -0.808144 q^{15} -4.99001 q^{16} +6.49256 q^{17} +13.1691 q^{18} +4.32335 q^{19} +0.222026 q^{20} +1.70296 q^{22} +3.15700 q^{23} +6.13665 q^{24} -4.93915 q^{25} +5.39972 q^{26} -15.5063 q^{27} +6.48417 q^{29} -1.37624 q^{30} -1.78122 q^{31} -4.75152 q^{32} -3.27614 q^{33} +11.0566 q^{34} +6.96033 q^{36} +8.38424 q^{37} +7.36248 q^{38} -10.3879 q^{39} -0.462057 q^{40} +0.553067 q^{41} +5.69023 q^{43} +0.900071 q^{44} +1.90756 q^{45} +5.37624 q^{46} -10.2971 q^{47} +16.3480 q^{48} -8.41117 q^{50} -21.2705 q^{51} +2.85393 q^{52} -10.1744 q^{53} -26.4065 q^{54} +0.246676 q^{55} -14.1639 q^{57} +11.0423 q^{58} +6.45660 q^{59} -0.727387 q^{60} +3.38149 q^{61} -3.03335 q^{62} +1.88838 q^{64} +0.782155 q^{65} -5.57913 q^{66} -3.65484 q^{67} +5.84377 q^{68} -10.3428 q^{69} +0.345158 q^{71} -14.4851 q^{72} -2.97942 q^{73} +14.2780 q^{74} +16.1813 q^{75} +3.89132 q^{76} -17.6902 q^{78} -3.77595 q^{79} -1.23091 q^{80} +27.6014 q^{81} +0.941851 q^{82} +6.34157 q^{83} +1.60156 q^{85} +9.69023 q^{86} -21.2430 q^{87} -1.87313 q^{88} +0.246676 q^{89} +3.24851 q^{90} +2.84152 q^{92} +5.83553 q^{93} -17.5356 q^{94} +1.06646 q^{95} +15.5667 q^{96} +4.81068 q^{97} +7.73309 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9}+O(q^{10}) $ 10 * q + 2 * q^2 + 18 * q^4 - 6 * q^8 + 22 * q^9	$ 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9} + 10 q^{11} + 8 q^{15} + 42 q^{16} + 6 q^{18} + 2 q^{22} + 4 q^{23} + 18 q^{25} + 12 q^{29} - 4 q^{30} - 30 q^{32} - 2 q^{36} + 40 q^{37} - 16 q^{39} - 8 q^{43} + 18 q^{44} + 44 q^{46} - 62 q^{50} + 16 q^{53} - 8 q^{57} - 28 q^{58} + 36 q^{60} + 106 q^{64} - 32 q^{65} - 4 q^{67} + 36 q^{71} - 90 q^{72} - 28 q^{74} - 112 q^{78} + 8 q^{79} - 6 q^{81} + 88 q^{85} + 32 q^{86} - 6 q^{88} - 52 q^{92} + 44 q^{93} - 64 q^{95} + 22 q^{99}+O(q^{100}) $ 10 * q + 2 * q^2 + 18 * q^4 - 6 * q^8 + 22 * q^9 + 10 * q^11 + 8 * q^15 + 42 * q^16 + 6 * q^18 + 2 * q^22 + 4 * q^23 + 18 * q^25 + 12 * q^29 - 4 * q^30 - 30 * q^32 - 2 * q^36 + 40 * q^37 - 16 * q^39 - 8 * q^43 + 18 * q^44 + 44 * q^46 - 62 * q^50 + 16 * q^53 - 8 * q^57 - 28 * q^58 + 36 * q^60 + 106 * q^64 - 32 * q^65 - 4 * q^67 + 36 * q^71 - 90 * q^72 - 28 * q^74 - 112 * q^78 + 8 * q^79 - 6 * q^81 + 88 * q^85 + 32 * q^86 - 6 * q^88 - 52 * q^92 + 44 * q^93 - 64 * q^95 + 22 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.70296	1.20417	0.602087	−	0.798430i	$-0.294336\pi$
$2$	1.70296	1.20417	0.602087	+	0.798430i	$0.294336\pi$
$3$	−3.27614	−1.89148	−0.945740	−	0.324924i	$-0.894661\pi$
$3$	−3.27614	−1.89148	−0.945740	+	0.324924i	$0.894661\pi$
$4$	0.900071	0.450036
$5$	0.246676	0.110317	0.0551584	−	0.998478i	$-0.482434\pi$
$5$	0.246676	0.110317	0.0551584	+	0.998478i	$0.482434\pi$
$6$	−5.57913	−2.27767
$7$	0	0
$7$	0	0
$8$	−1.87313	−0.662253
$9$	7.73309	2.57770
$10$	0.420079	0.132841
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−2.94876	−0.851233
$13$	3.17078	0.879417	0.439709	−	0.898140i	$-0.355082\pi$
$13$	3.17078	0.879417	0.439709	+	0.898140i	$0.355082\pi$
$14$	0	0
$15$	−0.808144	−0.208662
$16$	−4.99001	−1.24750
$17$	6.49256	1.57468	0.787339	−	0.616520i	$-0.211458\pi$
$17$	6.49256	1.57468	0.787339	+	0.616520i	$0.211458\pi$
$18$	13.1691	3.10400
$19$	4.32335	0.991844	0.495922	−	0.868367i	$-0.334830\pi$
$19$	4.32335	0.991844	0.495922	+	0.868367i	$0.334830\pi$
$20$	0.222026	0.0496464
$21$	0	0
$22$	1.70296	0.363072
$23$	3.15700	0.658279	0.329140	−	0.944281i	$-0.393241\pi$
$23$	3.15700	0.658279	0.329140	+	0.944281i	$0.393241\pi$
$24$	6.13665	1.25264
$25$	−4.93915	−0.987830
$26$	5.39972	1.05897
$27$	−15.5063	−2.98418
$28$	0	0
$29$	6.48417	1.20408	0.602040	−	0.798466i	$-0.294355\pi$
$29$	6.48417	1.20408	0.602040	+	0.798466i	$0.294355\pi$
$30$	−1.37624	−0.251265
$31$	−1.78122	−0.319917	−0.159958	−	0.987124i	$-0.551136\pi$
$31$	−1.78122	−0.319917	−0.159958	+	0.987124i	$0.551136\pi$
$32$	−4.75152	−0.839959
$33$	−3.27614	−0.570303
$34$	11.0566	1.89619
$35$	0	0
$36$	6.96033	1.16006
$37$	8.38424	1.37836	0.689180	−	0.724590i	$-0.257971\pi$
$37$	8.38424	1.37836	0.689180	+	0.724590i	$0.257971\pi$
$38$	7.36248	1.19435
$39$	−10.3879	−1.66340
$40$	−0.462057	−0.0730576
$41$	0.553067	0.0863746	0.0431873	−	0.999067i	$-0.486249\pi$
$41$	0.553067	0.0863746	0.0431873	+	0.999067i	$0.486249\pi$
$42$	0	0
$43$	5.69023	0.867752	0.433876	−	0.900973i	$-0.357146\pi$
$43$	5.69023	0.867752	0.433876	+	0.900973i	$0.357146\pi$
$44$	0.900071	0.135691
$45$	1.90756	0.284363
$46$	5.37624	0.792683
$47$	−10.2971	−1.50199	−0.750995	−	0.660308i	$-0.770426\pi$
$47$	−10.2971	−1.50199	−0.750995	+	0.660308i	$0.770426\pi$
$48$	16.3480	2.35963
$49$	0	0
$50$	−8.41117	−1.18952
$51$	−21.2705	−2.97847
$52$	2.85393	0.395769
$53$	−10.1744	−1.39756	−0.698780	−	0.715336i	$-0.746273\pi$
$53$	−10.1744	−1.39756	−0.698780	+	0.715336i	$0.746273\pi$
$54$	−26.4065	−3.59347
$55$	0.246676	0.0332617
$56$	0	0
$57$	−14.1639	−1.87605
$58$	11.0423	1.44992
$59$	6.45660	0.840577	0.420289	−	0.907390i	$-0.361929\pi$
$59$	6.45660	0.840577	0.420289	+	0.907390i	$0.361929\pi$
$60$	−0.727387	−0.0939053
$61$	3.38149	0.432956	0.216478	−	0.976287i	$-0.430543\pi$
$61$	3.38149	0.432956	0.216478	+	0.976287i	$0.430543\pi$
$62$	−3.03335	−0.385235
$63$	0	0
$64$	1.88838	0.236047
$65$	0.782155	0.0970144
$66$	−5.57913	−0.686744
$67$	−3.65484	−0.446510	−0.223255	−	0.974760i	$-0.571668\pi$
$67$	−3.65484	−0.446510	−0.223255	+	0.974760i	$0.571668\pi$
$68$	5.84377	0.708661
$69$	−10.3428	−1.24512
$70$	0	0
$71$	0.345158	0.0409627	0.0204813	−	0.999790i	$-0.493480\pi$
$71$	0.345158	0.0409627	0.0204813	+	0.999790i	$0.493480\pi$
$72$	−14.4851	−1.70709
$73$	−2.97942	−0.348715	−0.174357	−	0.984682i	$-0.555785\pi$
$73$	−2.97942	−0.348715	−0.174357	+	0.984682i	$0.555785\pi$
$74$	14.2780	1.65979
$75$	16.1813	1.86846
$76$	3.89132	0.446365
$77$	0	0
$78$	−17.6902	−2.00302
$79$	−3.77595	−0.424828	−0.212414	−	0.977180i	$-0.568132\pi$
$79$	−3.77595	−0.424828	−0.212414	+	0.977180i	$0.568132\pi$
$80$	−1.23091	−0.137620
$81$	27.6014	3.06682
$82$	0.941851	0.104010
$83$	6.34157	0.696078	0.348039	−	0.937480i	$-0.386848\pi$
$83$	6.34157	0.696078	0.348039	+	0.937480i	$0.386848\pi$
$84$	0	0
$85$	1.60156	0.173713
$86$	9.69023	1.04492
$87$	−21.2430	−2.27749
$88$	−1.87313	−0.199677
$89$	0.246676	0.0261476	0.0130738	−	0.999915i	$-0.495838\pi$
$89$	0.246676	0.0261476	0.0130738	+	0.999915i	$0.495838\pi$
$90$	3.24851	0.342423
$91$	0	0
$92$	2.84152	0.296249
$93$	5.83553	0.605116
$94$	−17.5356	−1.80866
$95$	1.06646	0.109417
$96$	15.5667	1.58877
$97$	4.81068	0.488451	0.244226	−	0.969718i	$-0.421466\pi$
$97$	4.81068	0.488451	0.244226	+	0.969718i	$0.421466\pi$
$98$	0	0
$99$	7.73309	0.777205
$100$	−4.44559	−0.444559
$101$	6.86468	0.683061	0.341531	−	0.939871i	$-0.389055\pi$
$101$	6.86468	0.683061	0.341531	+	0.939871i	$0.389055\pi$
$102$	−36.2229	−3.58660
$103$	−5.07835	−0.500385	−0.250192	−	0.968196i	$-0.580494\pi$
$103$	−5.07835	−0.500385	−0.250192	+	0.968196i	$0.580494\pi$
$104$	−5.93930	−0.582397
$105$	0	0
$106$	−17.3266	−1.68291
$107$	0.897562	0.0867706	0.0433853	−	0.999058i	$-0.486186\pi$
$107$	0.897562	0.0867706	0.0433853	+	0.999058i	$0.486186\pi$
$108$	−13.9567	−1.34299
$109$	5.71569	0.547464	0.273732	−	0.961806i	$-0.411742\pi$
$109$	5.71569	0.547464	0.273732	+	0.961806i	$0.411742\pi$
$110$	0.420079	0.0400529
$111$	−27.4679	−2.60714
$112$	0	0
$113$	−5.99001	−0.563493	−0.281747	−	0.959489i	$-0.590914\pi$
$113$	−5.99001	−0.563493	−0.281747	+	0.959489i	$0.590914\pi$
$114$	−24.1205	−2.25909
$115$	0.778754	0.0726192
$116$	5.83621	0.541879
$117$	24.5200	2.26687
$118$	10.9953	1.01220
$119$	0	0
$120$	1.51376	0.138187
$121$	1.00000	0.0909091
$122$	5.75855	0.521354
$123$	−1.81193	−0.163376
$124$	−1.60323	−0.143974
$125$	−2.45175	−0.219291
$126$	0	0
$127$	2.31399	0.205334	0.102667	−	0.994716i	$-0.467262\pi$
$127$	2.31399	0.205334	0.102667	+	0.994716i	$0.467262\pi$
$128$	12.7189	1.12420
$129$	−18.6420	−1.64133
$130$	1.33198	0.116822
$131$	−14.4800	−1.26513	−0.632564	−	0.774508i	$-0.717997\pi$
$131$	−14.4800	−1.26513	−0.632564	+	0.774508i	$0.717997\pi$
$132$	−2.94876	−0.256657
$133$	0	0
$134$	−6.22405	−0.537676
$135$	−3.82502	−0.329205
$136$	−12.1614	−1.04284
$137$	−10.9962	−0.939470	−0.469735	−	0.882807i	$-0.655651\pi$
$137$	−10.9962	−0.939470	−0.469735	+	0.882807i	$0.655651\pi$
$138$	−17.6133	−1.49934
$139$	−1.05087	−0.0891334	−0.0445667	−	0.999006i	$-0.514191\pi$
$139$	−1.05087	−0.0891334	−0.0445667	+	0.999006i	$0.514191\pi$
$140$	0	0
$141$	33.7348	2.84098
$142$	0.587790	0.0493262
$143$	3.17078	0.265154
$144$	−38.5882	−3.21569
$145$	1.59949	0.132830
$146$	−5.07383	−0.419913
$147$	0	0
$148$	7.54641	0.620311
$149$	18.4185	1.50890	0.754452	−	0.656356i	$-0.227903\pi$
$149$	18.4185	1.50890	0.754452	+	0.656356i	$0.227903\pi$
$150$	27.5562	2.24995
$151$	19.1564	1.55893	0.779463	−	0.626448i	$-0.215492\pi$
$151$	19.1564	1.55893	0.779463	+	0.626448i	$0.215492\pi$
$152$	−8.09821	−0.656851
$153$	50.2076	4.05904
$154$	0	0
$155$	−0.439384	−0.0352922
$156$	−9.34988	−0.748589
$157$	−23.0214	−1.83731	−0.918653	−	0.395065i	$-0.870722\pi$
$157$	−23.0214	−1.83731	−0.918653	+	0.395065i	$0.870722\pi$
$158$	−6.43029	−0.511567
$159$	33.3327	2.64346
$160$	−1.17209	−0.0926615
$161$	0	0
$162$	47.0041	3.69299
$163$	−4.93865	−0.386825	−0.193412	−	0.981118i	$-0.561956\pi$
$163$	−4.93865	−0.386825	−0.193412	+	0.981118i	$0.561956\pi$
$164$	0.497800	0.0388716
$165$	−0.808144	−0.0629139
$166$	10.7994	0.838199
$167$	−7.39244	−0.572044	−0.286022	−	0.958223i	$-0.592333\pi$
$167$	−7.39244	−0.572044	−0.286022	+	0.958223i	$0.592333\pi$
$168$	0	0
$169$	−2.94613	−0.226625
$170$	2.72739	0.209181
$171$	33.4328	2.55667
$172$	5.12161	0.390519
$173$	18.2383	1.38663	0.693316	−	0.720634i	$-0.256149\pi$
$173$	18.2383	1.38663	0.693316	+	0.720634i	$0.256149\pi$
$174$	−36.1760	−2.74250
$175$	0	0
$176$	−4.99001	−0.376136
$177$	−21.1527	−1.58994
$178$	0.420079	0.0314862
$179$	−2.37373	−0.177421	−0.0887103	−	0.996057i	$-0.528275\pi$
$179$	−2.37373	−0.177421	−0.0887103	+	0.996057i	$0.528275\pi$
$180$	1.71694	0.127973
$181$	21.6459	1.60893	0.804463	−	0.594002i	$-0.202453\pi$
$181$	21.6459	1.60893	0.804463	+	0.594002i	$0.202453\pi$
$182$	0	0
$183$	−11.0782	−0.818928
$184$	−5.91348	−0.435947
$185$	2.06819	0.152056
$186$	9.93767	0.728665
$187$	6.49256	0.474783
$188$	−9.26814	−0.675949
$189$	0	0
$190$	1.81615	0.131757
$191$	7.22154	0.522532	0.261266	−	0.965267i	$-0.415860\pi$
$191$	7.22154	0.522532	0.261266	+	0.965267i	$0.415860\pi$
$192$	−6.18658	−0.446478
$193$	−3.66604	−0.263887	−0.131944	−	0.991257i	$-0.542122\pi$
$193$	−3.66604	−0.263887	−0.131944	+	0.991257i	$0.542122\pi$
$194$	8.19240	0.588180
$195$	−2.56245	−0.183501
$196$	0	0
$197$	−3.81615	−0.271889	−0.135945	−	0.990716i	$-0.543407\pi$
$197$	−3.81615	−0.271889	−0.135945	+	0.990716i	$0.543407\pi$
$198$	13.1691	0.935890
$199$	19.3819	1.37395	0.686973	−	0.726683i	$-0.258939\pi$
$199$	19.3819	1.37395	0.686973	+	0.726683i	$0.258939\pi$
$200$	9.25169	0.654193
$201$	11.9738	0.844565
$202$	11.6903	0.822525
$203$	0	0
$204$	−19.1450	−1.34042
$205$	0.136428	0.00952856
$206$	−8.64822	−0.602550
$207$	24.4133	1.69684
$208$	−15.8223	−1.09708
$209$	4.32335	0.299052
$210$	0	0
$211$	24.1279	1.66103	0.830517	−	0.556993i	$-0.188045\pi$
$211$	24.1279	1.66103	0.830517	+	0.556993i	$0.188045\pi$
$212$	−9.15768	−0.628952
$213$	−1.13078	−0.0774801
$214$	1.52851	0.104487
$215$	1.40364	0.0957275
$216$	29.0453	1.97628
$217$	0	0
$218$	9.73359	0.659242
$219$	9.76100	0.659587
$220$	0.222026	0.0149690
$221$	20.5865	1.38480
$222$	−46.7768	−3.13945
$223$	−14.6952	−0.984061	−0.492030	−	0.870578i	$-0.663745\pi$
$223$	−14.6952	−0.984061	−0.492030	+	0.870578i	$0.663745\pi$
$224$	0	0
$225$	−38.1949	−2.54633
$226$	−10.2008	−0.678544
$227$	−9.19531	−0.610314	−0.305157	−	0.952302i	$-0.598709\pi$
$227$	−9.19531	−0.610314	−0.305157	+	0.952302i	$0.598709\pi$
$228$	−12.7485	−0.844290
$229$	−17.4339	−1.15207	−0.576033	−	0.817427i	$-0.695400\pi$
$229$	−17.4339	−1.15207	−0.576033	+	0.817427i	$0.695400\pi$
$230$	1.32619	0.0874461
$231$	0	0
$232$	−12.1457	−0.797405
$233$	−27.7505	−1.81799	−0.908997	−	0.416802i	$-0.863151\pi$
$233$	−27.7505	−1.81799	−0.908997	+	0.416802i	$0.863151\pi$
$234$	41.7565	2.72971
$235$	−2.54005	−0.165695
$236$	5.81140	0.378290
$237$	12.3705	0.803553
$238$	0	0
$239$	−22.5326	−1.45752	−0.728758	−	0.684772i	$-0.759902\pi$
$239$	−22.5326	−1.45752	−0.728758	+	0.684772i	$0.759902\pi$
$240$	4.03265	0.260306
$241$	7.22658	0.465505	0.232752	−	0.972536i	$-0.425227\pi$
$241$	7.22658	0.465505	0.232752	+	0.972536i	$0.425227\pi$
$242$	1.70296	0.109470
$243$	−43.9073	−2.81665
$244$	3.04359	0.194846
$245$	0	0
$246$	−3.08564	−0.196733
$247$	13.7084	0.872244
$248$	3.33647	0.211866
$249$	−20.7759	−1.31662
$250$	−4.17523	−0.264064
$251$	−12.8842	−0.813245	−0.406622	−	0.913596i	$-0.633294\pi$
$251$	−12.8842	−0.813245	−0.406622	+	0.913596i	$0.633294\pi$
$252$	0	0
$253$	3.15700	0.198479
$254$	3.94063	0.247257
$255$	−5.24692	−0.328575
$256$	17.8830	1.11769
$257$	−31.0932	−1.93954	−0.969771	−	0.244016i	$-0.921535\pi$
$257$	−31.0932	−1.93954	−0.969771	+	0.244016i	$0.921535\pi$
$258$	−31.7465	−1.97645
$259$	0	0
$260$	0.703995	0.0436599
$261$	50.1426	3.10375
$262$	−24.6589	−1.52343
$263$	−7.50215	−0.462603	−0.231301	−	0.972882i	$-0.574298\pi$
$263$	−7.50215	−0.462603	−0.231301	+	0.972882i	$0.574298\pi$
$264$	6.13665	0.377685
$265$	−2.50978	−0.154174
$266$	0	0
$267$	−0.808144	−0.0494576
$268$	−3.28962	−0.200945
$269$	1.93752	0.118133	0.0590663	−	0.998254i	$-0.481188\pi$
$269$	1.93752	0.118133	0.0590663	+	0.998254i	$0.481188\pi$
$270$	−6.51385	−0.396420
$271$	−22.9786	−1.39585	−0.697926	−	0.716170i	$-0.745893\pi$
$271$	−22.9786	−1.39585	−0.697926	+	0.716170i	$0.745893\pi$
$272$	−32.3980	−1.96442
$273$	0	0
$274$	−18.7261	−1.13129
$275$	−4.93915	−0.297842
$276$	−9.30922	−0.560349
$277$	−5.52095	−0.331722	−0.165861	−	0.986149i	$-0.553040\pi$
$277$	−5.52095	−0.331722	−0.165861	+	0.986149i	$0.553040\pi$
$278$	−1.78958	−0.107332
$279$	−13.7743	−0.824648
$280$	0	0
$281$	−7.28000	−0.434288	−0.217144	−	0.976140i	$-0.569674\pi$
$281$	−7.28000	−0.434288	−0.217144	+	0.976140i	$0.569674\pi$
$282$	57.4490	3.42104
$283$	13.2600	0.788225	0.394113	−	0.919062i	$-0.371052\pi$
$283$	13.2600	0.788225	0.394113	+	0.919062i	$0.371052\pi$
$284$	0.310666	0.0184347
$285$	−3.49389	−0.206960
$286$	5.39972	0.319292
$287$	0	0
$288$	−36.7440	−2.16516
$289$	25.1534	1.47961
$290$	2.72386	0.159951
$291$	−15.7605	−0.923895
$292$	−2.68169	−0.156934
$293$	5.56718	0.325238	0.162619	−	0.986689i	$-0.448006\pi$
$293$	5.56718	0.325238	0.162619	+	0.986689i	$0.448006\pi$
$294$	0	0
$295$	1.59269	0.0927297
$296$	−15.7048	−0.912823
$297$	−15.5063	−0.899765
$298$	31.3660	1.81698
$299$	10.0102	0.578902
$300$	14.5644	0.840874
$301$	0	0
$302$	32.6226	1.87722
$303$	−22.4897	−1.29200
$304$	−21.5736	−1.23733
$305$	0.834132	0.0477623
$306$	85.5015	4.88779
$307$	14.2467	0.813102	0.406551	−	0.913628i	$-0.366731\pi$
$307$	14.2467	0.813102	0.406551	+	0.913628i	$0.366731\pi$
$308$	0	0
$309$	16.6374	0.946468
$310$	−0.748253	−0.0424979
$311$	3.56834	0.202342	0.101171	−	0.994869i	$-0.467741\pi$
$311$	3.56834	0.202342	0.101171	+	0.994869i	$0.467741\pi$
$312$	19.4580	1.10159
$313$	−9.11394	−0.515150	−0.257575	−	0.966258i	$-0.582924\pi$
$313$	−9.11394	−0.515150	−0.257575	+	0.966258i	$0.582924\pi$
$314$	−39.2045	−2.21244
$315$	0	0
$316$	−3.39863	−0.191188
$317$	−11.7542	−0.660181	−0.330090	−	0.943949i	$-0.607079\pi$
$317$	−11.7542	−0.660181	−0.330090	+	0.943949i	$0.607079\pi$
$318$	56.7643	3.18318
$319$	6.48417	0.363044
$320$	0.465816	0.0260399
$321$	−2.94054	−0.164125
$322$	0	0
$323$	28.0696	1.56183
$324$	24.8432	1.38018
$325$	−15.6610	−0.868715
$326$	−8.41032	−0.465805
$327$	−18.7254	−1.03552
$328$	−1.03597	−0.0572018
$329$	0	0
$330$	−1.37624	−0.0757593
$331$	−25.6748	−1.41122	−0.705609	−	0.708602i	$-0.749326\pi$
$331$	−25.6748	−1.41122	−0.705609	+	0.708602i	$0.749326\pi$
$332$	5.70786	0.313260
$333$	64.8361	3.55299
$334$	−12.5890	−0.688841
$335$	−0.901561	−0.0492575
$336$	0	0
$337$	33.5644	1.82837	0.914185	−	0.405298i	$-0.132832\pi$
$337$	33.5644	1.82837	0.914185	+	0.405298i	$0.132832\pi$
$338$	−5.01713	−0.272896
$339$	19.6241	1.06584
$340$	1.44152	0.0781772
$341$	−1.78122	−0.0964585
$342$	56.9347	3.07868
$343$	0	0
$344$	−10.6586	−0.574671
$345$	−2.55131	−0.137358
$346$	31.0591	1.66975
$347$	−26.7567	−1.43637	−0.718187	−	0.695850i	$-0.755028\pi$
$347$	−26.7567	−1.43637	−0.718187	+	0.695850i	$0.755028\pi$
$348$	−19.1202	−1.02495
$349$	−19.9411	−1.06742	−0.533711	−	0.845667i	$-0.679203\pi$
$349$	−19.9411	−1.06742	−0.533711	+	0.845667i	$0.679203\pi$
$350$	0	0
$351$	−49.1670	−2.62434
$352$	−4.75152	−0.253257
$353$	−10.0436	−0.534566	−0.267283	−	0.963618i	$-0.586126\pi$
$353$	−10.0436	−0.534566	−0.267283	+	0.963618i	$0.586126\pi$
$354$	−36.0222	−1.91456
$355$	0.0851420	0.00451887
$356$	0.222026	0.0117673
$357$	0	0
$358$	−4.04236	−0.213645
$359$	−17.0519	−0.899964	−0.449982	−	0.893038i	$-0.648570\pi$
$359$	−17.0519	−0.899964	−0.449982	+	0.893038i	$0.648570\pi$
$360$	−3.57313	−0.188320
$361$	−0.308682	−0.0162464
$362$	36.8621	1.93743
$363$	−3.27614	−0.171953
$364$	0	0
$365$	−0.734951	−0.0384691
$366$	−18.8658	−0.986132
$367$	0.115028	0.00600443	0.00300221	−	0.999995i	$-0.499044\pi$
$367$	0.115028	0.00600443	0.00300221	+	0.999995i	$0.499044\pi$
$368$	−15.7535	−0.821206
$369$	4.27692	0.222648
$370$	3.52204	0.183102
$371$	0	0
$372$	5.25239	0.272324
$373$	−17.5106	−0.906664	−0.453332	−	0.891342i	$-0.649765\pi$
$373$	−17.5106	−0.906664	−0.453332	+	0.891342i	$0.649765\pi$
$374$	11.0566	0.571722
$375$	8.03226	0.414784
$376$	19.2879	0.994697
$377$	20.5599	1.05889
$378$	0	0
$379$	7.72561	0.396838	0.198419	−	0.980117i	$-0.436419\pi$
$379$	7.72561	0.396838	0.198419	+	0.980117i	$0.436419\pi$
$380$	0.959894	0.0492415
$381$	−7.58096	−0.388384
$382$	12.2980	0.629220
$383$	10.6044	0.541860	0.270930	−	0.962599i	$-0.412669\pi$
$383$	10.6044	0.541860	0.270930	+	0.962599i	$0.412669\pi$
$384$	−41.6688	−2.12640
$385$	0	0
$386$	−6.24311	−0.317766
$387$	44.0030	2.23680
$388$	4.32996	0.219820
$389$	−22.2172	−1.12646	−0.563228	−	0.826302i	$-0.690441\pi$
$389$	−22.2172	−1.12646	−0.563228	+	0.826302i	$0.690441\pi$
$390$	−4.36375	−0.220967
$391$	20.4970	1.03658
$392$	0	0
$393$	47.4386	2.39296
$394$	−6.49874	−0.327402
$395$	−0.931435	−0.0468656
$396$	6.96033	0.349770
$397$	−19.8781	−0.997654	−0.498827	−	0.866702i	$-0.666236\pi$
$397$	−19.8781	−0.997654	−0.498827	+	0.866702i	$0.666236\pi$
$398$	33.0066	1.65447
$399$	0	0
$400$	24.6464	1.23232
$401$	−24.7759	−1.23725	−0.618624	−	0.785687i	$-0.712309\pi$
$401$	−24.7759	−1.23725	−0.618624	+	0.785687i	$0.712309\pi$
$402$	20.3909	1.01700
$403$	−5.64787	−0.281340
$404$	6.17870	0.307402
$405$	6.80859	0.338322
$406$	0	0
$407$	8.38424	0.415591
$408$	39.8426	1.97250
$409$	21.7215	1.07406	0.537029	−	0.843564i	$-0.319546\pi$
$409$	21.7215	1.07406	0.537029	+	0.843564i	$0.319546\pi$
$410$	0.232332	0.0114740
$411$	36.0251	1.77699
$412$	−4.57088	−0.225191
$413$	0	0
$414$	41.5749	2.04330
$415$	1.56431	0.0767890
$416$	−15.0661	−0.738674
$417$	3.44279	0.168594
$418$	7.36248	0.360111
$419$	25.6815	1.25462	0.627311	−	0.778769i	$-0.284155\pi$
$419$	25.6815	1.25462	0.627311	+	0.778769i	$0.284155\pi$
$420$	0	0
$421$	28.7580	1.40158	0.700789	−	0.713369i	$-0.252832\pi$
$421$	28.7580	1.40158	0.700789	+	0.713369i	$0.252832\pi$
$422$	41.0889	2.00017
$423$	−79.6285	−3.87167
$424$	19.0580	0.925539
$425$	−32.0677	−1.55551
$426$	−1.92568	−0.0932995
$427$	0	0
$428$	0.807869	0.0390498
$429$	−10.3879	−0.501534
$430$	2.39034	0.115273
$431$	−9.84034	−0.473992	−0.236996	−	0.971511i	$-0.576163\pi$
$431$	−9.84034	−0.473992	−0.236996	+	0.971511i	$0.576163\pi$
$432$	77.3765	3.72278
$433$	0.424750	0.0204122	0.0102061	−	0.999948i	$-0.496751\pi$
$433$	0.424750	0.0204122	0.0102061	+	0.999948i	$0.496751\pi$
$434$	0	0
$435$	−5.24014	−0.251245
$436$	5.14453	0.246378
$437$	13.6488	0.652910
$438$	16.6226	0.794258
$439$	3.98983	0.190424	0.0952121	−	0.995457i	$-0.469647\pi$
$439$	3.98983	0.190424	0.0952121	+	0.995457i	$0.469647\pi$
$440$	−0.462057	−0.0220277
$441$	0	0
$442$	35.0580	1.66754
$443$	−19.9380	−0.947285	−0.473642	−	0.880717i	$-0.657061\pi$
$443$	−19.9380	−0.947285	−0.473642	+	0.880717i	$0.657061\pi$
$444$	−24.7231	−1.17331
$445$	0.0608489	0.00288451
$446$	−25.0253	−1.18498
$447$	−60.3416	−2.85406
$448$	0	0
$449$	9.97100	0.470561	0.235280	−	0.971928i	$-0.424399\pi$
$449$	9.97100	0.470561	0.235280	+	0.971928i	$0.424399\pi$
$450$	−65.0444	−3.06622
$451$	0.553067	0.0260429
$452$	−5.39144	−0.253592
$453$	−62.7591	−2.94868
$454$	−15.6592	−0.734925
$455$	0	0
$456$	26.5309	1.24242
$457$	34.9248	1.63371	0.816856	−	0.576841i	$-0.195715\pi$
$457$	34.9248	1.63371	0.816856	+	0.576841i	$0.195715\pi$
$458$	−29.6893	−1.38729
$459$	−100.675	−4.69912
$460$	0.700934	0.0326812
$461$	−26.8042	−1.24839	−0.624197	−	0.781267i	$-0.714574\pi$
$461$	−26.8042	−1.24839	−0.624197	+	0.781267i	$0.714574\pi$
$462$	0	0
$463$	−28.1253	−1.30709	−0.653547	−	0.756886i	$-0.726720\pi$
$463$	−28.1253	−1.30709	−0.653547	+	0.756886i	$0.726720\pi$
$464$	−32.3561	−1.50209
$465$	1.43948	0.0667544
$466$	−47.2580	−2.18918
$467$	27.7857	1.28577	0.642884	−	0.765964i	$-0.277738\pi$
$467$	27.7857	1.28577	0.642884	+	0.765964i	$0.277738\pi$
$468$	22.0697	1.02017
$469$	0	0
$470$	−4.32560	−0.199525
$471$	75.4213	3.47523
$472$	−12.0941	−0.556675
$473$	5.69023	0.261637
$474$	21.0665	0.967618
$475$	−21.3537	−0.979773
$476$	0	0
$477$	−78.6795	−3.60249
$478$	−38.3722	−1.75510
$479$	9.27456	0.423766	0.211883	−	0.977295i	$-0.432040\pi$
$479$	9.27456	0.423766	0.211883	+	0.977295i	$0.432040\pi$
$480$	3.83991	0.175267
$481$	26.5846	1.21215
$482$	12.3066	0.560549
$483$	0	0
$484$	0.900071	0.0409123
$485$	1.18668	0.0538843
$486$	−74.7723	−3.39174
$487$	10.6356	0.481944	0.240972	−	0.970532i	$-0.422534\pi$
$487$	10.6356	0.481944	0.240972	+	0.970532i	$0.422534\pi$
$488$	−6.33399	−0.286726
$489$	16.1797	0.731672
$490$	0	0
$491$	−19.9832	−0.901829	−0.450915	−	0.892567i	$-0.648902\pi$
$491$	−19.9832	−0.901829	−0.450915	+	0.892567i	$0.648902\pi$
$492$	−1.63086	−0.0735249
$493$	42.0989	1.89604
$494$	23.3448	1.05033
$495$	1.90756	0.0857387
$496$	8.88832	0.399097
$497$	0	0
$498$	−35.3805	−1.58544
$499$	−10.9536	−0.490351	−0.245175	−	0.969479i	$-0.578846\pi$
$499$	−10.9536	−0.490351	−0.245175	+	0.969479i	$0.578846\pi$
$500$	−2.20675	−0.0986887
$501$	24.2187	1.08201
$502$	−21.9413	−0.979288
$503$	25.5069	1.13729	0.568647	−	0.822582i	$-0.307467\pi$
$503$	25.5069	1.13729	0.568647	+	0.822582i	$0.307467\pi$
$504$	0	0
$505$	1.69335	0.0753531
$506$	5.37624	0.239003
$507$	9.65192	0.428657
$508$	2.08276	0.0924074
$509$	−5.16877	−0.229102	−0.114551	−	0.993417i	$-0.536543\pi$
$509$	−5.16877	−0.229102	−0.114551	+	0.993417i	$0.536543\pi$
$510$	−8.93530	−0.395662
$511$	0	0
$512$	5.01624	0.221689
$513$	−67.0389	−2.95984
$514$	−52.9505	−2.33555
$515$	−1.25271	−0.0552008
$516$	−16.7791	−0.738659
$517$	−10.2971	−0.452867
$518$	0	0
$519$	−59.7512	−2.62279
$520$	−1.46508	−0.0642481
$521$	−15.4837	−0.678352	−0.339176	−	0.940723i	$-0.610148\pi$
$521$	−15.4837	−0.678352	−0.339176	+	0.940723i	$0.610148\pi$
$522$	85.3909	3.73746
$523$	10.2761	0.449344	0.224672	−	0.974434i	$-0.427869\pi$
$523$	10.2761	0.449344	0.224672	+	0.974434i	$0.427869\pi$
$524$	−13.0331	−0.569352
$525$	0	0
$526$	−12.7759	−0.557054
$527$	−11.5647	−0.503766
$528$	16.3480	0.711455
$529$	−13.0334	−0.566669
$530$	−4.27405	−0.185653
$531$	49.9294	2.16675
$532$	0	0
$533$	1.75366	0.0759593
$534$	−1.37624	−0.0595556
$535$	0.221407	0.00957224
$536$	6.84601	0.295703
$537$	7.77666	0.335588
$538$	3.29952	0.142252
$539$	0	0
$540$	−3.44279	−0.148154
$541$	32.5212	1.39820	0.699099	−	0.715025i	$-0.253585\pi$
$541$	32.5212	1.39820	0.699099	+	0.715025i	$0.253585\pi$
$542$	−39.1317	−1.68085
$543$	−70.9150	−3.04325
$544$	−30.8496	−1.32266
$545$	1.40992	0.0603944
$546$	0	0
$547$	−32.7759	−1.40139	−0.700697	−	0.713459i	$-0.747128\pi$
$547$	−32.7759	−1.40139	−0.700697	+	0.713459i	$0.747128\pi$
$548$	−9.89738	−0.422795
$549$	26.1494	1.11603
$550$	−8.41117	−0.358654
$551$	28.0333	1.19426
$552$	19.3734	0.824586
$553$	0	0
$554$	−9.40195	−0.399451
$555$	−6.77567	−0.287611
$556$	−0.945855	−0.0401132
$557$	−14.3574	−0.608341	−0.304170	−	0.952618i	$-0.598379\pi$
$557$	−14.3574	−0.608341	−0.304170	+	0.952618i	$0.598379\pi$
$558$	−23.4571	−0.993020
$559$	18.0425	0.763116
$560$	0	0
$561$	−21.2705	−0.898043
$562$	−12.3975	−0.522959
$563$	13.3422	0.562305	0.281152	−	0.959663i	$-0.409283\pi$
$563$	13.3422	0.562305	0.281152	+	0.959663i	$0.409283\pi$
$564$	30.3637	1.27854
$565$	−1.47759	−0.0621627
$566$	22.5812	0.949160
$567$	0	0
$568$	−0.646527	−0.0271277
$569$	−18.6491	−0.781809	−0.390904	−	0.920431i	$-0.627838\pi$
$569$	−18.6491	−0.781809	−0.390904	+	0.920431i	$0.627838\pi$
$570$	−5.94994	−0.249216
$571$	32.0508	1.34128	0.670642	−	0.741781i	$-0.266019\pi$
$571$	32.0508	1.34128	0.670642	+	0.741781i	$0.266019\pi$
$572$	2.85393	0.119329
$573$	−23.6588	−0.988359
$574$	0	0
$575$	−15.5929	−0.650268
$576$	14.6030	0.608457
$577$	16.4352	0.684206	0.342103	−	0.939662i	$-0.388861\pi$
$577$	16.4352	0.684206	0.342103	+	0.939662i	$0.388861\pi$
$578$	42.8352	1.78171
$579$	12.0104	0.499137
$580$	1.43965	0.0597783
$581$	0	0
$582$	−26.8394	−1.11253
$583$	−10.1744	−0.421380
$584$	5.58086	0.230937
$585$	6.04848	0.250074
$586$	9.48069	0.391644
$587$	−11.8148	−0.487647	−0.243824	−	0.969820i	$-0.578402\pi$
$587$	−11.8148	−0.487647	−0.243824	+	0.969820i	$0.578402\pi$
$588$	0	0
$589$	−7.70083	−0.317307
$590$	2.71228	0.111663
$591$	12.5022	0.514273
$592$	−41.8375	−1.71951
$593$	20.7452	0.851905	0.425952	−	0.904746i	$-0.359939\pi$
$593$	20.7452	0.851905	0.425952	+	0.904746i	$0.359939\pi$
$594$	−26.4065	−1.08347
$595$	0	0
$596$	16.5780	0.679060
$597$	−63.4978	−2.59879
$598$	17.0469	0.697099
$599$	8.71038	0.355897	0.177948	−	0.984040i	$-0.443054\pi$
$599$	8.71038	0.355897	0.177948	+	0.984040i	$0.443054\pi$
$600$	−30.3098	−1.23739
$601$	−46.2310	−1.88580	−0.942902	−	0.333072i	$-0.891915\pi$
$601$	−46.2310	−1.88580	−0.942902	+	0.333072i	$0.891915\pi$
$602$	0	0
$603$	−28.2632	−1.15097
$604$	17.2421	0.701572
$605$	0.246676	0.0100288
$606$	−38.2990	−1.55579
$607$	30.1304	1.22295	0.611477	−	0.791262i	$-0.290576\pi$
$607$	30.1304	1.22295	0.611477	+	0.791262i	$0.290576\pi$
$608$	−20.5425	−0.833108
$609$	0	0
$610$	1.42049	0.0575141
$611$	−32.6499	−1.32088
$612$	45.1904	1.82671
$613$	41.7643	1.68684	0.843421	−	0.537253i	$-0.180538\pi$
$613$	41.7643	1.68684	0.843421	+	0.537253i	$0.180538\pi$
$614$	24.2616	0.979117
$615$	−0.446958	−0.0180231
$616$	0	0
$617$	8.94182	0.359984	0.179992	−	0.983668i	$-0.442393\pi$
$617$	8.94182	0.359984	0.179992	+	0.983668i	$0.442393\pi$
$618$	28.3328	1.13971
$619$	30.9126	1.24248	0.621241	−	0.783620i	$-0.286629\pi$
$619$	30.9126	1.24248	0.621241	+	0.783620i	$0.286629\pi$
$620$	−0.395477	−0.0158827
$621$	−48.9532	−1.96442
$622$	6.07674	0.243655
$623$	0	0
$624$	51.8359	2.07510
$625$	24.0910	0.963639
$626$	−15.5207	−0.620331
$627$	−14.1639	−0.565651
$628$	−20.7209	−0.826853
$629$	54.4352	2.17047
$630$	0	0
$631$	30.5300	1.21538	0.607691	−	0.794174i	$-0.292096\pi$
$631$	30.5300	1.21538	0.607691	+	0.794174i	$0.292096\pi$
$632$	7.07286	0.281343
$633$	−79.0464	−3.14181
$634$	−20.0169	−0.794973
$635$	0.570805	0.0226517
$636$	30.0018	1.18965
$637$	0	0
$638$	11.0423	0.437168
$639$	2.66914	0.105589
$640$	3.13744	0.124018
$641$	−10.4287	−0.411907	−0.205953	−	0.978562i	$-0.566030\pi$
$641$	−10.4287	−0.411907	−0.205953	+	0.978562i	$0.566030\pi$
$642$	−5.00762	−0.197635
$643$	16.4973	0.650590	0.325295	−	0.945613i	$-0.394536\pi$
$643$	16.4973	0.650590	0.325295	+	0.945613i	$0.394536\pi$
$644$	0	0
$645$	−4.59852	−0.181067
$646$	47.8014	1.88072
$647$	−41.0615	−1.61429	−0.807147	−	0.590350i	$-0.798990\pi$
$647$	−41.0615	−1.61429	−0.807147	+	0.590350i	$0.798990\pi$
$648$	−51.7011	−2.03101
$649$	6.45660	0.253444
$650$	−26.6700	−1.04608
$651$	0	0
$652$	−4.44514	−0.174085
$653$	−20.1454	−0.788350	−0.394175	−	0.919035i	$-0.628970\pi$
$653$	−20.1454	−0.788350	−0.394175	+	0.919035i	$0.628970\pi$
$654$	−31.8886	−1.24694
$655$	−3.57187	−0.139565
$656$	−2.75981	−0.107753
$657$	−23.0401	−0.898881
$658$	0	0
$659$	6.46125	0.251694	0.125847	−	0.992050i	$-0.459835\pi$
$659$	6.46125	0.251694	0.125847	+	0.992050i	$0.459835\pi$
$660$	−0.727387	−0.0283135
$661$	−14.3775	−0.559221	−0.279611	−	0.960114i	$-0.590205\pi$
$661$	−14.3775	−0.559221	−0.279611	+	0.960114i	$0.590205\pi$
$662$	−43.7232	−1.69935
$663$	−67.4443	−2.61932
$664$	−11.8786	−0.460979
$665$	0	0
$666$	110.413	4.27842
$667$	20.4705	0.792620
$668$	−6.65372	−0.257440
$669$	48.1434	1.86133
$670$	−1.53532	−0.0593146
$671$	3.38149	0.130541
$672$	0	0
$673$	27.0790	1.04382	0.521909	−	0.853001i	$-0.325220\pi$
$673$	27.0790	1.04382	0.521909	+	0.853001i	$0.325220\pi$
$674$	57.1588	2.20168
$675$	76.5878	2.94786
$676$	−2.65172	−0.101989
$677$	−15.8329	−0.608506	−0.304253	−	0.952591i	$-0.598407\pi$
$677$	−15.8329	−0.608506	−0.304253	+	0.952591i	$0.598407\pi$
$678$	33.4191	1.28345
$679$	0	0
$680$	−2.99993	−0.115042
$681$	30.1251	1.15440
$682$	−3.03335	−0.116153
$683$	37.0041	1.41592	0.707962	−	0.706251i	$-0.249615\pi$
$683$	37.0041	1.41592	0.707962	+	0.706251i	$0.249615\pi$
$684$	30.0919	1.15059
$685$	−2.71250	−0.103639
$686$	0	0
$687$	57.1159	2.17911
$688$	−28.3943	−1.08252
$689$	−32.2608	−1.22904
$690$	−4.34477	−0.165403
$691$	−14.5966	−0.555283	−0.277641	−	0.960685i	$-0.589553\pi$
$691$	−14.5966	−0.555283	−0.277641	+	0.960685i	$0.589553\pi$
$692$	16.4158	0.624034
$693$	0	0
$694$	−45.5656	−1.72965
$695$	−0.259223	−0.00983290
$696$	39.7911	1.50828
$697$	3.59082	0.136012
$698$	−33.9589	−1.28536
$699$	90.9145	3.43870
$700$	0	0
$701$	−40.3475	−1.52390	−0.761952	−	0.647633i	$-0.775759\pi$
$701$	−40.3475	−1.52390	−0.761952	+	0.647633i	$0.775759\pi$
$702$	−83.7294	−3.16016
$703$	36.2480	1.36712
$704$	1.88838	0.0711708
$705$	8.32155	0.313408
$706$	−17.1038	−0.643710
$707$	0	0
$708$	−19.0389	−0.715527
$709$	0.763572	0.0286766	0.0143383	−	0.999897i	$-0.495436\pi$
$709$	0.763572	0.0286766	0.0143383	+	0.999897i	$0.495436\pi$
$710$	0.144993	0.00544150
$711$	−29.1998	−1.09508
$712$	−0.462057	−0.0173163
$713$	−5.62331	−0.210594
$714$	0	0
$715$	0.782155	0.0292509
$716$	−2.13652	−0.0798456
$717$	73.8201	2.75686
$718$	−29.0387	−1.08371
$719$	7.59172	0.283123	0.141562	−	0.989929i	$-0.454788\pi$
$719$	7.59172	0.283123	0.141562	+	0.989929i	$0.454788\pi$
$720$	−9.51878	−0.354744
$721$	0	0
$722$	−0.525673	−0.0195635
$723$	−23.6753	−0.880493
$724$	19.4828	0.724074
$725$	−32.0263	−1.18943
$726$	−5.57913	−0.207061
$727$	−2.08478	−0.0773204	−0.0386602	−	0.999252i	$-0.512309\pi$
$727$	−2.08478	−0.0773204	−0.0386602	+	0.999252i	$0.512309\pi$
$728$	0	0
$729$	61.0421	2.26082
$730$	−1.25159	−0.0463235
$731$	36.9442	1.36643
$732$	−9.97121	−0.368547
$733$	37.7859	1.39566	0.697828	−	0.716266i	$-0.254150\pi$
$733$	37.7859	1.39566	0.697828	+	0.716266i	$0.254150\pi$
$734$	0.195889	0.00723038
$735$	0	0
$736$	−15.0005	−0.552927
$737$	−3.65484	−0.134628
$738$	7.28342	0.268106
$739$	36.8466	1.35543	0.677713	−	0.735327i	$-0.262971\pi$
$739$	36.8466	1.35543	0.677713	+	0.735327i	$0.262971\pi$
$740$	1.86152	0.0684307
$741$	−44.9106	−1.64983
$742$	0	0
$743$	−44.4863	−1.63204	−0.816022	−	0.578020i	$-0.803825\pi$
$743$	−44.4863	−1.63204	−0.816022	+	0.578020i	$0.803825\pi$
$744$	−10.9307	−0.400740
$745$	4.54340	0.166457
$746$	−29.8198	−1.09178
$747$	49.0399	1.79428
$748$	5.84377	0.213669
$749$	0	0
$750$	13.6786	0.499473
$751$	26.6146	0.971179	0.485590	−	0.874187i	$-0.338605\pi$
$751$	26.6146	0.971179	0.485590	+	0.874187i	$0.338605\pi$
$752$	51.3828	1.87374
$753$	42.2105	1.53824
$754$	35.0127	1.27509
$755$	4.72542	0.171976
$756$	0	0
$757$	−6.38731	−0.232151	−0.116075	−	0.993240i	$-0.537031\pi$
$757$	−6.38731	−0.232151	−0.116075	+	0.993240i	$0.537031\pi$
$758$	13.1564	0.477862
$759$	−10.3428	−0.375418
$760$	−1.99763	−0.0724617
$761$	−40.1614	−1.45585	−0.727925	−	0.685657i	$-0.759515\pi$
$761$	−40.1614	−1.45585	−0.727925	+	0.685657i	$0.759515\pi$
$762$	−12.9101	−0.467682
$763$	0	0
$764$	6.49990	0.235158
$765$	12.3850	0.447780
$766$	18.0589	0.652494
$767$	20.4725	0.739218
$768$	−58.5871	−2.11408
$769$	0.481136	0.0173502	0.00867510	−	0.999962i	$-0.497239\pi$
$769$	0.481136	0.0173502	0.00867510	+	0.999962i	$0.497239\pi$
$770$	0	0
$771$	101.866	3.66861
$772$	−3.29969	−0.118759
$773$	−24.1864	−0.869924	−0.434962	−	0.900449i	$-0.643238\pi$
$773$	−24.1864	−0.869924	−0.434962	+	0.900449i	$0.643238\pi$
$774$	74.9354	2.69350
$775$	8.79772	0.316023
$776$	−9.01106	−0.323478
$777$	0	0
$778$	−37.8349	−1.35645
$779$	2.39110	0.0856701
$780$	−2.30639	−0.0825819
$781$	0.345158	0.0123507
$782$	34.9056	1.24822
$783$	−100.545	−3.59319
$784$	0	0
$785$	−5.67881	−0.202686
$786$	80.7861	2.88154
$787$	−37.8595	−1.34954	−0.674772	−	0.738026i	$-0.735758\pi$
$787$	−37.8595	−1.34954	−0.674772	+	0.738026i	$0.735758\pi$
$788$	−3.43480	−0.122360
$789$	24.5781	0.875004
$790$	−1.58620	−0.0564343
$791$	0	0
$792$	−14.4851	−0.514706
$793$	10.7220	0.380749
$794$	−33.8516	−1.20135
$795$	8.22237	0.291618
$796$	17.4451	0.618325
$797$	27.0930	0.959683	0.479842	−	0.877355i	$-0.340694\pi$
$797$	27.0930	0.959683	0.479842	+	0.877355i	$0.340694\pi$
$798$	0	0
$799$	−66.8547	−2.36515
$800$	23.4685	0.829737
$801$	1.90756	0.0674005
$802$	−42.1923	−1.48986
$803$	−2.97942	−0.105141
$804$	10.7772	0.380084
$805$	0	0
$806$	−9.61809	−0.338783
$807$	−6.34758	−0.223445
$808$	−12.8585	−0.452359
$809$	15.7254	0.552876	0.276438	−	0.961032i	$-0.410846\pi$
$809$	15.7254	0.552876	0.276438	+	0.961032i	$0.410846\pi$
$810$	11.5948	0.407398
$811$	−32.3327	−1.13536	−0.567678	−	0.823251i	$-0.692158\pi$
$811$	−32.3327	−1.13536	−0.567678	+	0.823251i	$0.692158\pi$
$812$	0	0
$813$	75.2811	2.64023
$814$	14.2780	0.500444
$815$	−1.21824	−0.0426732
$816$	106.140	3.71565
$817$	24.6008	0.860674
$818$	36.9908	1.29335
$819$	0	0
$820$	0.122795	0.00428819
$821$	9.55190	0.333364	0.166682	−	0.986011i	$-0.446695\pi$
$821$	9.55190	0.333364	0.166682	+	0.986011i	$0.446695\pi$
$822$	61.3494	2.13980
$823$	28.6991	1.00039	0.500194	−	0.865913i	$-0.333262\pi$
$823$	28.6991	1.00039	0.500194	+	0.865913i	$0.333262\pi$
$824$	9.51243	0.331381
$825$	16.1813	0.563362
$826$	0	0
$827$	−9.66064	−0.335933	−0.167967	−	0.985793i	$-0.553720\pi$
$827$	−9.66064	−0.335933	−0.167967	+	0.985793i	$0.553720\pi$
$828$	21.9737	0.763640
$829$	−16.4010	−0.569629	−0.284815	−	0.958583i	$-0.591932\pi$
$829$	−16.4010	−0.569629	−0.284815	+	0.958583i	$0.591932\pi$
$830$	2.66396	0.0924673
$831$	18.0874	0.627445
$832$	5.98763	0.207584
$833$	0	0
$834$	5.86293	0.203017
$835$	−1.82353	−0.0631060
$836$	3.89132	0.134584
$837$	27.6201	0.954690
$838$	43.7345	1.51078
$839$	15.0128	0.518301	0.259150	−	0.965837i	$-0.416557\pi$
$839$	15.0128	0.518301	0.259150	+	0.965837i	$0.416557\pi$
$840$	0	0
$841$	13.0444	0.449807
$842$	48.9736	1.68774
$843$	23.8503	0.821448
$844$	21.7168	0.747524
$845$	−0.726738	−0.0250005
$846$	−135.604	−4.66217
$847$	0	0
$848$	50.7704	1.74346
$849$	−43.4416	−1.49091
$850$	−54.6101	−1.87311
$851$	26.4690	0.907346
$852$	−1.01779	−0.0348688
$853$	−57.4026	−1.96543	−0.982714	−	0.185133i	$-0.940729\pi$
$853$	−57.4026	−1.96543	−0.982714	+	0.185133i	$0.940729\pi$
$854$	0	0
$855$	8.24706	0.282044
$856$	−1.68125	−0.0574641
$857$	−32.1817	−1.09931	−0.549654	−	0.835393i	$-0.685240\pi$
$857$	−32.1817	−1.09931	−0.549654	+	0.835393i	$0.685240\pi$
$858$	−17.6902	−0.603934
$859$	−23.9856	−0.818380	−0.409190	−	0.912449i	$-0.634189\pi$
$859$	−23.9856	−0.818380	−0.409190	+	0.912449i	$0.634189\pi$
$860$	1.26338	0.0430808
$861$	0	0
$862$	−16.7577	−0.570769
$863$	36.7485	1.25093	0.625467	−	0.780251i	$-0.284909\pi$
$863$	36.7485	1.25093	0.625467	+	0.780251i	$0.284909\pi$
$864$	73.6784	2.50659
$865$	4.49894	0.152969
$866$	0.723331	0.0245798
$867$	−82.4060	−2.79865
$868$	0	0
$869$	−3.77595	−0.128090
$870$	−8.92374	−0.302543
$871$	−11.5887	−0.392669
$872$	−10.7063	−0.362560
$873$	37.2015	1.25908
$874$	23.2433	0.786217
$875$	0	0
$876$	8.78559	0.296838
$877$	−25.3446	−0.855826	−0.427913	−	0.903820i	$-0.640751\pi$
$877$	−25.3446	−0.855826	−0.427913	+	0.903820i	$0.640751\pi$
$878$	6.79452	0.229304
$879$	−18.2389	−0.615182
$880$	−1.23091	−0.0414941
$881$	12.7302	0.428890	0.214445	−	0.976736i	$-0.431206\pi$
$881$	12.7302	0.428890	0.214445	+	0.976736i	$0.431206\pi$
$882$	0	0
$883$	10.0508	0.338235	0.169117	−	0.985596i	$-0.445908\pi$
$883$	10.0508	0.338235	0.169117	+	0.985596i	$0.445908\pi$
$884$	18.5293	0.623209
$885$	−5.21786	−0.175396
$886$	−33.9537	−1.14070
$887$	18.9974	0.637871	0.318936	−	0.947776i	$-0.396675\pi$
$887$	18.9974	0.637871	0.318936	+	0.947776i	$0.396675\pi$
$888$	51.4511	1.72659
$889$	0	0
$890$	0.103623	0.00347346
$891$	27.6014	0.924682
$892$	−13.2267	−0.442862
$893$	−44.5180	−1.48974
$894$	−102.759	−3.43679
$895$	−0.585541	−0.0195725
$896$	0	0
$897$	−32.7947	−1.09498
$898$	16.9802	0.566637
$899$	−11.5497	−0.385205
$900$	−34.3781	−1.14594
$901$	−66.0579	−2.20071
$902$	0.941851	0.0313602
$903$	0	0
$904$	11.2201	0.373175
$905$	5.33951	0.177491
$906$	−106.876	−3.55072
$907$	−59.1419	−1.96378	−0.981888	−	0.189464i	$-0.939325\pi$
$907$	−59.1419	−1.96378	−0.981888	+	0.189464i	$0.939325\pi$
$908$	−8.27643	−0.274663
$909$	53.0852	1.76073
$910$	0	0
$911$	−11.5307	−0.382028	−0.191014	−	0.981587i	$-0.561178\pi$
$911$	−11.5307	−0.382028	−0.191014	+	0.981587i	$0.561178\pi$
$912$	70.6780	2.34038
$913$	6.34157	0.209875
$914$	59.4755	1.96728
$915$	−2.73273	−0.0903414
$916$	−15.6918	−0.518471
$917$	0	0
$918$	−171.446	−5.65856
$919$	27.8583	0.918962	0.459481	−	0.888188i	$-0.348036\pi$
$919$	27.8583	0.918962	0.459481	+	0.888188i	$0.348036\pi$
$920$	−1.45871	−0.0480923
$921$	−46.6742	−1.53797
$922$	−45.6464	−1.50328
$923$	1.09442	0.0360233
$924$	0	0
$925$	−41.4110	−1.36159
$926$	−47.8963	−1.57397
$927$	−39.2713	−1.28984
$928$	−30.8097	−1.01138
$929$	51.3944	1.68619	0.843097	−	0.537761i	$-0.180730\pi$
$929$	51.3944	1.68619	0.843097	+	0.537761i	$0.180730\pi$
$930$	2.45138	0.0803839
$931$	0	0
$932$	−24.9774	−0.818162
$933$	−11.6904	−0.382726
$934$	47.3179	1.54829
$935$	1.60156	0.0523765
$936$	−45.9292	−1.50124
$937$	−8.07437	−0.263778	−0.131889	−	0.991264i	$-0.542104\pi$
$937$	−8.07437	−0.263778	−0.131889	+	0.991264i	$0.542104\pi$
$938$	0	0
$939$	29.8585	0.974397
$940$	−2.28622	−0.0745684
$941$	24.3020	0.792223	0.396112	−	0.918202i	$-0.370359\pi$
$941$	24.3020	0.792223	0.396112	+	0.918202i	$0.370359\pi$
$942$	128.439	4.18478
$943$	1.74603	0.0568586
$944$	−32.2185	−1.04862
$945$	0	0
$946$	9.69023	0.315056
$947$	40.5660	1.31822	0.659109	−	0.752047i	$-0.270933\pi$
$947$	40.5660	1.31822	0.659109	+	0.752047i	$0.270933\pi$
$948$	11.1344	0.361628
$949$	−9.44710	−0.306666
$950$	−36.3644	−1.17982
$951$	38.5083	1.24872
$952$	0	0
$953$	−4.52347	−0.146529	−0.0732647	−	0.997313i	$-0.523342\pi$
$953$	−4.52347	−0.146529	−0.0732647	+	0.997313i	$0.523342\pi$
$954$	−133.988	−4.33802
$955$	1.78138	0.0576440
$956$	−20.2810	−0.655934
$957$	−21.2430	−0.686690
$958$	15.7942	0.510288
$959$	0	0
$960$	−1.52608	−0.0492540
$961$	−27.8273	−0.897653
$962$	45.2725	1.45964
$963$	6.94092	0.223668
$964$	6.50443	0.209494
$965$	−0.904322	−0.0291112
$966$	0	0
$967$	−48.3508	−1.55486	−0.777428	−	0.628972i	$-0.783476\pi$
$967$	−48.3508	−1.55486	−0.777428	+	0.628972i	$0.783476\pi$
$968$	−1.87313	−0.0602048
$969$	−91.9599	−2.95418
$970$	2.02087	0.0648861
$971$	−12.7515	−0.409217	−0.204608	−	0.978844i	$-0.565592\pi$
$971$	−12.7515	−0.409217	−0.204608	+	0.978844i	$0.565592\pi$
$972$	−39.5197	−1.26759
$973$	0	0
$974$	18.1120	0.580345
$975$	51.3076	1.64316
$976$	−16.8737	−0.540114
$977$	21.9973	0.703757	0.351879	−	0.936046i	$-0.385543\pi$
$977$	21.9973	0.703757	0.351879	+	0.936046i	$0.385543\pi$
$978$	27.5534	0.881060
$979$	0.246676	0.00788379
$980$	0	0
$981$	44.2000	1.41120
$982$	−34.0306	−1.08596
$983$	51.4813	1.64200	0.820999	−	0.570930i	$-0.193417\pi$
$983$	51.4813	1.64200	0.820999	+	0.570930i	$0.193417\pi$
$984$	3.39398	0.108196
$985$	−0.941350	−0.0299939
$986$	71.6927	2.28316
$987$	0	0
$988$	12.3385	0.392541
$989$	17.9640	0.571223
$990$	3.24851	0.103244
$991$	−12.8667	−0.408724	−0.204362	−	0.978895i	$-0.565512\pi$
$991$	−12.8667	−0.408724	−0.204362	+	0.978895i	$0.565512\pi$
$992$	8.46351	0.268717
$993$	84.1144	2.66929
$994$	0	0
$995$	4.78104	0.151569
$996$	−18.6998	−0.592524
$997$	30.0438	0.951496	0.475748	−	0.879582i	$-0.342177\pi$
$997$	30.0438	0.951496	0.475748	+	0.879582i	$0.342177\pi$
$998$	−18.6535	−0.590468
$999$	−130.008	−4.11328

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.a.l.1.7	✓	10
3.2	odd	2		4851.2.a.cg.1.3		10
4.3	odd	2		8624.2.a.df.1.10		10
7.2	even	3		539.2.e.o.67.4		20
7.3	odd	6		539.2.e.o.177.3		20
7.4	even	3		539.2.e.o.177.4		20
7.5	odd	6		539.2.e.o.67.3		20
7.6	odd	2	inner	539.2.a.l.1.8	yes	10
11.10	odd	2		5929.2.a.bv.1.3		10
21.20	even	2		4851.2.a.cg.1.4		10
28.27	even	2		8624.2.a.df.1.1		10
77.76	even	2		5929.2.a.bv.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.a.l.1.7	✓	10	1.1	even	1	trivial
539.2.a.l.1.8	yes	10	7.6	odd	2	inner
539.2.e.o.67.3		20	7.5	odd	6
539.2.e.o.67.4		20	7.2	even	3
539.2.e.o.177.3		20	7.3	odd	6
539.2.e.o.177.4		20	7.4	even	3
4851.2.a.cg.1.3		10	3.2	odd	2
4851.2.a.cg.1.4		10	21.20	even	2
5929.2.a.bv.1.3		10	11.10	odd	2
5929.2.a.bv.1.4		10	77.76	even	2
8624.2.a.df.1.1		10	28.27	even	2
8624.2.a.df.1.10		10	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)

\(f(q)\)	\(=\)	\(q+1.70296 q^{2} -3.27614 q^{3} +0.900071 q^{4} +0.246676 q^{5} -5.57913 q^{6} -1.87313 q^{8} +7.73309 q^{9} +O(q^{10})\)	\(q+1.70296 q^{2} -3.27614 q^{3} +0.900071 q^{4} +0.246676 q^{5} -5.57913 q^{6} -1.87313 q^{8} +7.73309 q^{9} +0.420079 q^{10} +1.00000 q^{11} -2.94876 q^{12} +3.17078 q^{13} -0.808144 q^{15} -4.99001 q^{16} +6.49256 q^{17} +13.1691 q^{18} +4.32335 q^{19} +0.222026 q^{20} +1.70296 q^{22} +3.15700 q^{23} +6.13665 q^{24} -4.93915 q^{25} +5.39972 q^{26} -15.5063 q^{27} +6.48417 q^{29} -1.37624 q^{30} -1.78122 q^{31} -4.75152 q^{32} -3.27614 q^{33} +11.0566 q^{34} +6.96033 q^{36} +8.38424 q^{37} +7.36248 q^{38} -10.3879 q^{39} -0.462057 q^{40} +0.553067 q^{41} +5.69023 q^{43} +0.900071 q^{44} +1.90756 q^{45} +5.37624 q^{46} -10.2971 q^{47} +16.3480 q^{48} -8.41117 q^{50} -21.2705 q^{51} +2.85393 q^{52} -10.1744 q^{53} -26.4065 q^{54} +0.246676 q^{55} -14.1639 q^{57} +11.0423 q^{58} +6.45660 q^{59} -0.727387 q^{60} +3.38149 q^{61} -3.03335 q^{62} +1.88838 q^{64} +0.782155 q^{65} -5.57913 q^{66} -3.65484 q^{67} +5.84377 q^{68} -10.3428 q^{69} +0.345158 q^{71} -14.4851 q^{72} -2.97942 q^{73} +14.2780 q^{74} +16.1813 q^{75} +3.89132 q^{76} -17.6902 q^{78} -3.77595 q^{79} -1.23091 q^{80} +27.6014 q^{81} +0.941851 q^{82} +6.34157 q^{83} +1.60156 q^{85} +9.69023 q^{86} -21.2430 q^{87} -1.87313 q^{88} +0.246676 q^{89} +3.24851 q^{90} +2.84152 q^{92} +5.83553 q^{93} -17.5356 q^{94} +1.06646 q^{95} +15.5667 q^{96} +4.81068 q^{97} +7.73309 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9}+O(q^{10}) \) 10 * q + 2 * q^2 + 18 * q^4 - 6 * q^8 + 22 * q^9	\( 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9} + 10 q^{11} + 8 q^{15} + 42 q^{16} + 6 q^{18} + 2 q^{22} + 4 q^{23} + 18 q^{25} + 12 q^{29} - 4 q^{30} - 30 q^{32} - 2 q^{36} + 40 q^{37} - 16 q^{39} - 8 q^{43} + 18 q^{44} + 44 q^{46} - 62 q^{50} + 16 q^{53} - 8 q^{57} - 28 q^{58} + 36 q^{60} + 106 q^{64} - 32 q^{65} - 4 q^{67} + 36 q^{71} - 90 q^{72} - 28 q^{74} - 112 q^{78} + 8 q^{79} - 6 q^{81} + 88 q^{85} + 32 q^{86} - 6 q^{88} - 52 q^{92} + 44 q^{93} - 64 q^{95} + 22 q^{99}+O(q^{100}) \) 10 * q + 2 * q^2 + 18 * q^4 - 6 * q^8 + 22 * q^9 + 10 * q^11 + 8 * q^15 + 42 * q^16 + 6 * q^18 + 2 * q^22 + 4 * q^23 + 18 * q^25 + 12 * q^29 - 4 * q^30 - 30 * q^32 - 2 * q^36 + 40 * q^37 - 16 * q^39 - 8 * q^43 + 18 * q^44 + 44 * q^46 - 62 * q^50 + 16 * q^53 - 8 * q^57 - 28 * q^58 + 36 * q^60 + 106 * q^64 - 32 * q^65 - 4 * q^67 + 36 * q^71 - 90 * q^72 - 28 * q^74 - 112 * q^78 + 8 * q^79 - 6 * q^81 + 88 * q^85 + 32 * q^86 - 6 * q^88 - 52 * q^92 + 44 * q^93 - 64 * q^95 + 22 * q^99

Introduction

L-functions

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