LMFDB - Embedded newform 8450.2.a.bz.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

sage:traces = [3,3,-1,3,0,-1,5,3,-4,0,-4,-1,0,5,0,3,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.4735897080$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1690)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	8450.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} -0.445042 q^{6} +3.24698 q^{7} +1.00000 q^{8} -2.80194 q^{9} +1.60388 q^{11} -0.445042 q^{12} +3.24698 q^{14} +1.00000 q^{16} -3.10992 q^{17} -2.80194 q^{18} -2.89008 q^{19} -1.44504 q^{21} +1.60388 q^{22} -6.78986 q^{23} -0.445042 q^{24} +2.58211 q^{27} +3.24698 q^{28} +4.04892 q^{29} -8.31767 q^{31} +1.00000 q^{32} -0.713792 q^{33} -3.10992 q^{34} -2.80194 q^{36} -3.20775 q^{37} -2.89008 q^{38} -10.6746 q^{41} -1.44504 q^{42} +5.18598 q^{43} +1.60388 q^{44} -6.78986 q^{46} +3.97823 q^{47} -0.445042 q^{48} +3.54288 q^{49} +1.38404 q^{51} +11.0858 q^{53} +2.58211 q^{54} +3.24698 q^{56} +1.28621 q^{57} +4.04892 q^{58} -5.28621 q^{59} -13.1250 q^{61} -8.31767 q^{62} -9.09783 q^{63} +1.00000 q^{64} -0.713792 q^{66} +12.3937 q^{67} -3.10992 q^{68} +3.02177 q^{69} -7.50604 q^{71} -2.80194 q^{72} -12.2741 q^{73} -3.20775 q^{74} -2.89008 q^{76} +5.20775 q^{77} -11.3056 q^{79} +7.25667 q^{81} -10.6746 q^{82} -10.4209 q^{83} -1.44504 q^{84} +5.18598 q^{86} -1.80194 q^{87} +1.60388 q^{88} -11.3817 q^{89} -6.78986 q^{92} +3.70171 q^{93} +3.97823 q^{94} -0.445042 q^{96} -6.05429 q^{97} +3.54288 q^{98} -4.49396 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} + 5 q^{7} + 3 q^{8} - 4 q^{9} - 4 q^{11} - q^{12} + 5 q^{14} + 3 q^{16} - 10 q^{17} - 4 q^{18} - 8 q^{19} - 4 q^{21} - 4 q^{22} + 3 q^{23} - q^{24} + 2 q^{27}+ \cdots - 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^6 + 5 * q^7 + 3 * q^8 - 4 * q^9 - 4 * q^11 - q^12 + 5 * q^14 + 3 * q^16 - 10 * q^17 - 4 * q^18 - 8 * q^19 - 4 * q^21 - 4 * q^22 + 3 * q^23 - q^24 + 2 * q^27 + 5 * q^28 + 3 * q^29 - 8 * q^31 + 3 * q^32 + 6 * q^33 - 10 * q^34 - 4 * q^36 + 8 * q^37 - 8 * q^38 - 11 * q^41 - 4 * q^42 + q^43 - 4 * q^44 + 3 * q^46 + 15 * q^47 - q^48 - 8 * q^49 - 6 * q^51 - 4 * q^53 + 2 * q^54 + 5 * q^56 + 12 * q^57 + 3 * q^58 - 24 * q^59 - 15 * q^61 - 8 * q^62 - 9 * q^63 + 3 * q^64 + 6 * q^66 + 5 * q^67 - 10 * q^68 + 6 * q^69 - 32 * q^71 - 4 * q^72 - 26 * q^73 + 8 * q^74 - 8 * q^76 - 2 * q^77 + 2 * q^79 - 5 * q^81 - 11 * q^82 + 7 * q^83 - 4 * q^84 + q^86 - q^87 - 4 * q^88 + 17 * q^89 + 3 * q^92 - 16 * q^93 + 15 * q^94 - q^96 - 6 * q^97 - 8 * q^98 - 4 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−0.445042	−0.256945	−0.128473	−	0.991713i	$-0.541007\pi$
$3$	−0.445042	−0.256945	−0.128473	+	0.991713i	$0.541007\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−0.445042	−0.181688
$7$	3.24698	1.22724	0.613621	−	0.789600i	$-0.289712\pi$
$7$	3.24698	1.22724	0.613621	+	0.789600i	$0.289712\pi$
$8$	1.00000	0.353553
$9$	−2.80194	−0.933979
$10$	0	0
$11$	1.60388	0.483587	0.241793	−	0.970328i	$-0.422264\pi$
$11$	1.60388	0.483587	0.241793	+	0.970328i	$0.422264\pi$
$12$	−0.445042	−0.128473
$13$	0	0
$13$	0	0
$14$	3.24698	0.867792
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.10992	−0.754265	−0.377133	−	0.926159i	$-0.623090\pi$
$17$	−3.10992	−0.754265	−0.377133	+	0.926159i	$0.623090\pi$
$18$	−2.80194	−0.660423
$19$	−2.89008	−0.663031	−0.331515	−	0.943450i	$-0.607560\pi$
$19$	−2.89008	−0.663031	−0.331515	+	0.943450i	$0.607560\pi$
$20$	0	0
$21$	−1.44504	−0.315334
$22$	1.60388	0.341947
$23$	−6.78986	−1.41578	−0.707891	−	0.706321i	$-0.750353\pi$
$23$	−6.78986	−1.41578	−0.707891	+	0.706321i	$0.750353\pi$
$24$	−0.445042	−0.0908438
$25$	0	0
$26$	0	0
$27$	2.58211	0.496926
$28$	3.24698	0.613621
$29$	4.04892	0.751865	0.375933	−	0.926647i	$-0.377322\pi$
$29$	4.04892	0.751865	0.375933	+	0.926647i	$0.377322\pi$
$30$	0	0
$31$	−8.31767	−1.49390	−0.746949	−	0.664882i	$-0.768482\pi$
$31$	−8.31767	−1.49390	−0.746949	+	0.664882i	$0.768482\pi$
$32$	1.00000	0.176777
$33$	−0.713792	−0.124255
$34$	−3.10992	−0.533346
$35$	0	0
$36$	−2.80194	−0.466990
$37$	−3.20775	−0.527351	−0.263676	−	0.964611i	$-0.584935\pi$
$37$	−3.20775	−0.527351	−0.263676	+	0.964611i	$0.584935\pi$
$38$	−2.89008	−0.468833
$39$	0	0
$40$	0	0
$41$	−10.6746	−1.66709	−0.833543	−	0.552454i	$-0.813691\pi$
$41$	−10.6746	−1.66709	−0.833543	+	0.552454i	$0.813691\pi$
$42$	−1.44504	−0.222975
$43$	5.18598	0.790855	0.395427	−	0.918497i	$-0.370597\pi$
$43$	5.18598	0.790855	0.395427	+	0.918497i	$0.370597\pi$
$44$	1.60388	0.241793
$45$	0	0
$46$	−6.78986	−1.00111
$47$	3.97823	0.580284	0.290142	−	0.956984i	$-0.406297\pi$
$47$	3.97823	0.580284	0.290142	+	0.956984i	$0.406297\pi$
$48$	−0.445042	−0.0642363
$49$	3.54288	0.506125
$50$	0	0
$51$	1.38404	0.193805
$52$	0	0
$53$	11.0858	1.52275	0.761373	−	0.648314i	$-0.224526\pi$
$53$	11.0858	1.52275	0.761373	+	0.648314i	$0.224526\pi$
$54$	2.58211	0.351380
$55$	0	0
$56$	3.24698	0.433896
$57$	1.28621	0.170362
$58$	4.04892	0.531649
$59$	−5.28621	−0.688206	−0.344103	−	0.938932i	$-0.611817\pi$
$59$	−5.28621	−0.688206	−0.344103	+	0.938932i	$0.611817\pi$
$60$	0	0
$61$	−13.1250	−1.68048	−0.840241	−	0.542213i	$-0.817586\pi$
$61$	−13.1250	−1.68048	−0.840241	+	0.542213i	$0.817586\pi$
$62$	−8.31767	−1.05634
$63$	−9.09783	−1.14622
$64$	1.00000	0.125000
$65$	0	0
$66$	−0.713792	−0.0878617
$67$	12.3937	1.51414	0.757068	−	0.653336i	$-0.226631\pi$
$67$	12.3937	1.51414	0.757068	+	0.653336i	$0.226631\pi$
$68$	−3.10992	−0.377133
$69$	3.02177	0.363778
$70$	0	0
$71$	−7.50604	−0.890803	−0.445402	−	0.895331i	$-0.646939\pi$
$71$	−7.50604	−0.890803	−0.445402	+	0.895331i	$0.646939\pi$
$72$	−2.80194	−0.330212
$73$	−12.2741	−1.43658	−0.718289	−	0.695745i	$-0.755074\pi$
$73$	−12.2741	−1.43658	−0.718289	+	0.695745i	$0.755074\pi$
$74$	−3.20775	−0.372893
$75$	0	0
$76$	−2.89008	−0.331515
$77$	5.20775	0.593478
$78$	0	0
$79$	−11.3056	−1.27198	−0.635989	−	0.771698i	$-0.719408\pi$
$79$	−11.3056	−1.27198	−0.635989	+	0.771698i	$0.719408\pi$
$80$	0	0
$81$	7.25667	0.806296
$82$	−10.6746	−1.17881
$83$	−10.4209	−1.14384	−0.571920	−	0.820309i	$-0.693801\pi$
$83$	−10.4209	−1.14384	−0.571920	+	0.820309i	$0.693801\pi$
$84$	−1.44504	−0.157667
$85$	0	0
$86$	5.18598	0.559219
$87$	−1.80194	−0.193188
$88$	1.60388	0.170974
$89$	−11.3817	−1.20645	−0.603226	−	0.797570i	$-0.706118\pi$
$89$	−11.3817	−1.20645	−0.603226	+	0.797570i	$0.706118\pi$
$90$	0	0
$91$	0	0
$92$	−6.78986	−0.707891
$93$	3.70171	0.383849
$94$	3.97823	0.410323
$95$	0	0
$96$	−0.445042	−0.0454219
$97$	−6.05429	−0.614720	−0.307360	−	0.951593i	$-0.599446\pi$
$97$	−6.05429	−0.614720	−0.307360	+	0.951593i	$0.599446\pi$
$98$	3.54288	0.357885
$99$	−4.49396	−0.451660
$100$	0	0
$101$	−4.19806	−0.417723	−0.208861	−	0.977945i	$-0.566976\pi$
$101$	−4.19806	−0.417723	−0.208861	+	0.977945i	$0.566976\pi$
$102$	1.38404	0.137041
$103$	−11.5211	−1.13521	−0.567604	−	0.823302i	$-0.692130\pi$
$103$	−11.5211	−1.13521	−0.567604	+	0.823302i	$0.692130\pi$
$104$	0	0
$105$	0	0
$106$	11.0858	1.07674
$107$	4.43967	0.429199	0.214599	−	0.976702i	$-0.431155\pi$
$107$	4.43967	0.429199	0.214599	+	0.976702i	$0.431155\pi$
$108$	2.58211	0.248463
$109$	10.5526	1.01075	0.505376	−	0.862899i	$-0.331354\pi$
$109$	10.5526	1.01075	0.505376	+	0.862899i	$0.331354\pi$
$110$	0	0
$111$	1.42758	0.135500
$112$	3.24698	0.306811
$113$	−4.31767	−0.406172	−0.203086	−	0.979161i	$-0.565097\pi$
$113$	−4.31767	−0.406172	−0.203086	+	0.979161i	$0.565097\pi$
$114$	1.28621	0.120464
$115$	0	0
$116$	4.04892	0.375933
$117$	0	0
$118$	−5.28621	−0.486635
$119$	−10.0978	−0.925667
$120$	0	0
$121$	−8.42758	−0.766144
$122$	−13.1250	−1.18828
$123$	4.75063	0.428350
$124$	−8.31767	−0.746949
$125$	0	0
$126$	−9.09783	−0.810500
$127$	2.80731	0.249109	0.124554	−	0.992213i	$-0.460250\pi$
$127$	2.80731	0.249109	0.124554	+	0.992213i	$0.460250\pi$
$128$	1.00000	0.0883883
$129$	−2.30798	−0.203206
$130$	0	0
$131$	2.37196	0.207239	0.103620	−	0.994617i	$-0.466958\pi$
$131$	2.37196	0.207239	0.103620	+	0.994617i	$0.466958\pi$
$132$	−0.713792	−0.0621276
$133$	−9.38404	−0.813700
$134$	12.3937	1.07066
$135$	0	0
$136$	−3.10992	−0.266673
$137$	16.8552	1.44003	0.720017	−	0.693956i	$-0.244134\pi$
$137$	16.8552	1.44003	0.720017	+	0.693956i	$0.244134\pi$
$138$	3.02177	0.257230
$139$	5.97584	0.506864	0.253432	−	0.967353i	$-0.418441\pi$
$139$	5.97584	0.506864	0.253432	+	0.967353i	$0.418441\pi$
$140$	0	0
$141$	−1.77048	−0.149101
$142$	−7.50604	−0.629893
$143$	0	0
$144$	−2.80194	−0.233495
$145$	0	0
$146$	−12.2741	−1.01581
$147$	−1.57673	−0.130046
$148$	−3.20775	−0.263676
$149$	14.9148	1.22187	0.610936	−	0.791680i	$-0.290793\pi$
$149$	14.9148	1.22187	0.610936	+	0.791680i	$0.290793\pi$
$150$	0	0
$151$	9.56033	0.778009	0.389005	−	0.921236i	$-0.372819\pi$
$151$	9.56033	0.778009	0.389005	+	0.921236i	$0.372819\pi$
$152$	−2.89008	−0.234417
$153$	8.71379	0.704468
$154$	5.20775	0.419653
$155$	0	0
$156$	0	0
$157$	13.7560	1.09785	0.548924	−	0.835872i	$-0.315038\pi$
$157$	13.7560	1.09785	0.548924	+	0.835872i	$0.315038\pi$
$158$	−11.3056	−0.899424
$159$	−4.93362	−0.391262
$160$	0	0
$161$	−22.0465	−1.73751
$162$	7.25667	0.570138
$163$	13.5308	1.05981	0.529907	−	0.848056i	$-0.322227\pi$
$163$	13.5308	1.05981	0.529907	+	0.848056i	$0.322227\pi$
$164$	−10.6746	−0.833543
$165$	0	0
$166$	−10.4209	−0.808817
$167$	15.1685	1.17378	0.586888	−	0.809668i	$-0.300353\pi$
$167$	15.1685	1.17378	0.586888	+	0.809668i	$0.300353\pi$
$168$	−1.44504	−0.111487
$169$	0	0
$170$	0	0
$171$	8.09783	0.619257
$172$	5.18598	0.395427
$173$	−4.27413	−0.324956	−0.162478	−	0.986712i	$-0.551949\pi$
$173$	−4.27413	−0.324956	−0.162478	+	0.986712i	$0.551949\pi$
$174$	−1.80194	−0.136605
$175$	0	0
$176$	1.60388	0.120897
$177$	2.35258	0.176831
$178$	−11.3817	−0.853091
$179$	15.6233	1.16774	0.583868	−	0.811848i	$-0.301538\pi$
$179$	15.6233	1.16774	0.583868	+	0.811848i	$0.301538\pi$
$180$	0	0
$181$	16.1933	1.20364	0.601818	−	0.798633i	$-0.294443\pi$
$181$	16.1933	1.20364	0.601818	+	0.798633i	$0.294443\pi$
$182$	0	0
$183$	5.84117	0.431791
$184$	−6.78986	−0.500555
$185$	0	0
$186$	3.70171	0.271423
$187$	−4.98792	−0.364753
$188$	3.97823	0.290142
$189$	8.38404	0.609849
$190$	0	0
$191$	−26.5676	−1.92237	−0.961183	−	0.275911i	$-0.911020\pi$
$191$	−26.5676	−1.92237	−0.961183	+	0.275911i	$0.911020\pi$
$192$	−0.445042	−0.0321181
$193$	−13.6582	−0.983137	−0.491568	−	0.870839i	$-0.663576\pi$
$193$	−13.6582	−0.983137	−0.491568	+	0.870839i	$0.663576\pi$
$194$	−6.05429	−0.434673
$195$	0	0
$196$	3.54288	0.253063
$197$	14.0978	1.00443	0.502215	−	0.864743i	$-0.332519\pi$
$197$	14.0978	1.00443	0.502215	+	0.864743i	$0.332519\pi$
$198$	−4.49396	−0.319372
$199$	−21.1293	−1.49782	−0.748908	−	0.662674i	$-0.769422\pi$
$199$	−21.1293	−1.49782	−0.748908	+	0.662674i	$0.769422\pi$
$200$	0	0
$201$	−5.51573	−0.389050
$202$	−4.19806	−0.295375
$203$	13.1468	0.922721
$204$	1.38404	0.0969024
$205$	0	0
$206$	−11.5211	−0.802714
$207$	19.0248	1.32231
$208$	0	0
$209$	−4.63533	−0.320633
$210$	0	0
$211$	−18.3720	−1.26478	−0.632389	−	0.774651i	$-0.717926\pi$
$211$	−18.3720	−1.26478	−0.632389	+	0.774651i	$0.717926\pi$
$212$	11.0858	0.761373
$213$	3.34050	0.228887
$214$	4.43967	0.303489
$215$	0	0
$216$	2.58211	0.175690
$217$	−27.0073	−1.83337
$218$	10.5526	0.714710
$219$	5.46250	0.369122
$220$	0	0
$221$	0	0
$222$	1.42758	0.0958131
$223$	−14.2687	−0.955506	−0.477753	−	0.878494i	$-0.658549\pi$
$223$	−14.2687	−0.955506	−0.477753	+	0.878494i	$0.658549\pi$
$224$	3.24698	0.216948
$225$	0	0
$226$	−4.31767	−0.287207
$227$	−27.4959	−1.82497	−0.912483	−	0.409115i	$-0.865837\pi$
$227$	−27.4959	−1.82497	−0.912483	+	0.409115i	$0.865837\pi$
$228$	1.28621	0.0851812
$229$	19.6963	1.30157	0.650785	−	0.759262i	$-0.274440\pi$
$229$	19.6963	1.30157	0.650785	+	0.759262i	$0.274440\pi$
$230$	0	0
$231$	−2.31767	−0.152491
$232$	4.04892	0.265824
$233$	23.5362	1.54191	0.770953	−	0.636892i	$-0.219780\pi$
$233$	23.5362	1.54191	0.770953	+	0.636892i	$0.219780\pi$
$234$	0	0
$235$	0	0
$236$	−5.28621	−0.344103
$237$	5.03146	0.326828
$238$	−10.0978	−0.654545
$239$	−14.4155	−0.932461	−0.466231	−	0.884663i	$-0.654388\pi$
$239$	−14.4155	−0.932461	−0.466231	+	0.884663i	$0.654388\pi$
$240$	0	0
$241$	15.6692	1.00934	0.504671	−	0.863312i	$-0.331614\pi$
$241$	15.6692	1.00934	0.504671	+	0.863312i	$0.331614\pi$
$242$	−8.42758	−0.541746
$243$	−10.9758	−0.704100
$244$	−13.1250	−0.840241
$245$	0	0
$246$	4.75063	0.302889
$247$	0	0
$248$	−8.31767	−0.528172
$249$	4.63773	0.293904
$250$	0	0
$251$	−15.1535	−0.956478	−0.478239	−	0.878230i	$-0.658725\pi$
$251$	−15.1535	−0.956478	−0.478239	+	0.878230i	$0.658725\pi$
$252$	−9.09783	−0.573110
$253$	−10.8901	−0.684654
$254$	2.80731	0.176147
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.8552	−1.17615	−0.588076	−	0.808805i	$-0.700115\pi$
$257$	−18.8552	−1.17615	−0.588076	+	0.808805i	$0.700115\pi$
$258$	−2.30798	−0.143688
$259$	−10.4155	−0.647188
$260$	0	0
$261$	−11.3448	−0.702226
$262$	2.37196	0.146540
$263$	2.18060	0.134462	0.0672309	−	0.997737i	$-0.478584\pi$
$263$	2.18060	0.134462	0.0672309	+	0.997737i	$0.478584\pi$
$264$	−0.713792	−0.0439308
$265$	0	0
$266$	−9.38404	−0.575373
$267$	5.06531	0.309992
$268$	12.3937	0.757068
$269$	9.73125	0.593325	0.296662	−	0.954982i	$-0.404126\pi$
$269$	9.73125	0.593325	0.296662	+	0.954982i	$0.404126\pi$
$270$	0	0
$271$	4.05429	0.246281	0.123140	−	0.992389i	$-0.460703\pi$
$271$	4.05429	0.246281	0.123140	+	0.992389i	$0.460703\pi$
$272$	−3.10992	−0.188566
$273$	0	0
$274$	16.8552	1.01826
$275$	0	0
$276$	3.02177	0.181889
$277$	−30.0301	−1.80434	−0.902168	−	0.431385i	$-0.858025\pi$
$277$	−30.0301	−1.80434	−0.902168	+	0.431385i	$0.858025\pi$
$278$	5.97584	0.358407
$279$	23.3056	1.39527
$280$	0	0
$281$	22.2640	1.32816	0.664078	−	0.747663i	$-0.268824\pi$
$281$	22.2640	1.32816	0.664078	+	0.747663i	$0.268824\pi$
$282$	−1.77048	−0.105430
$283$	8.88471	0.528141	0.264071	−	0.964503i	$-0.414935\pi$
$283$	8.88471	0.528141	0.264071	+	0.964503i	$0.414935\pi$
$284$	−7.50604	−0.445402
$285$	0	0
$286$	0	0
$287$	−34.6601	−2.04592
$288$	−2.80194	−0.165106
$289$	−7.32842	−0.431084
$290$	0	0
$291$	2.69441	0.157949
$292$	−12.2741	−0.718289
$293$	−21.5362	−1.25816	−0.629078	−	0.777342i	$-0.716568\pi$
$293$	−21.5362	−1.25816	−0.629078	+	0.777342i	$0.716568\pi$
$294$	−1.57673	−0.0919567
$295$	0	0
$296$	−3.20775	−0.186447
$297$	4.14138	0.240307
$298$	14.9148	0.863993
$299$	0	0
$300$	0	0
$301$	16.8388	0.970571
$302$	9.56033	0.550135
$303$	1.86831	0.107332
$304$	−2.89008	−0.165758
$305$	0	0
$306$	8.71379	0.498134
$307$	10.3230	0.589167	0.294584	−	0.955626i	$-0.404819\pi$
$307$	10.3230	0.589167	0.294584	+	0.955626i	$0.404819\pi$
$308$	5.20775	0.296739
$309$	5.12737	0.291686
$310$	0	0
$311$	7.50125	0.425357	0.212679	−	0.977122i	$-0.431781\pi$
$311$	7.50125	0.425357	0.212679	+	0.977122i	$0.431781\pi$
$312$	0	0
$313$	−5.60388	−0.316750	−0.158375	−	0.987379i	$-0.550625\pi$
$313$	−5.60388	−0.316750	−0.158375	+	0.987379i	$0.550625\pi$
$314$	13.7560	0.776296
$315$	0	0
$316$	−11.3056	−0.635989
$317$	11.7802	0.661640	0.330820	−	0.943694i	$-0.392675\pi$
$317$	11.7802	0.661640	0.330820	+	0.943694i	$0.392675\pi$
$318$	−4.93362	−0.276664
$319$	6.49396	0.363592
$320$	0	0
$321$	−1.97584	−0.110280
$322$	−22.0465	−1.22860
$323$	8.98792	0.500101
$324$	7.25667	0.403148
$325$	0	0
$326$	13.5308	0.749401
$327$	−4.69633	−0.259708
$328$	−10.6746	−0.589404
$329$	12.9172	0.712150
$330$	0	0
$331$	−6.46980	−0.355612	−0.177806	−	0.984066i	$-0.556900\pi$
$331$	−6.46980	−0.355612	−0.177806	+	0.984066i	$0.556900\pi$
$332$	−10.4209	−0.571920
$333$	8.98792	0.492535
$334$	15.1685	0.829985
$335$	0	0
$336$	−1.44504	−0.0788335
$337$	5.35988	0.291971	0.145986	−	0.989287i	$-0.453365\pi$
$337$	5.35988	0.291971	0.145986	+	0.989287i	$0.453365\pi$
$338$	0	0
$339$	1.92154	0.104364
$340$	0	0
$341$	−13.3405	−0.722429
$342$	8.09783	0.437881
$343$	−11.2252	−0.606104
$344$	5.18598	0.279609
$345$	0	0
$346$	−4.27413	−0.229778
$347$	−3.56571	−0.191417	−0.0957087	−	0.995409i	$-0.530512\pi$
$347$	−3.56571	−0.191417	−0.0957087	+	0.995409i	$0.530512\pi$
$348$	−1.80194	−0.0965940
$349$	−7.97584	−0.426937	−0.213468	−	0.976950i	$-0.568476\pi$
$349$	−7.97584	−0.426937	−0.213468	+	0.976950i	$0.568476\pi$
$350$	0	0
$351$	0	0
$352$	1.60388	0.0854868
$353$	1.60388	0.0853657	0.0426828	−	0.999089i	$-0.486410\pi$
$353$	1.60388	0.0853657	0.0426828	+	0.999089i	$0.486410\pi$
$354$	2.35258	0.125038
$355$	0	0
$356$	−11.3817	−0.603226
$357$	4.49396	0.237846
$358$	15.6233	0.825715
$359$	−8.60255	−0.454025	−0.227013	−	0.973892i	$-0.572896\pi$
$359$	−8.60255	−0.454025	−0.227013	+	0.973892i	$0.572896\pi$
$360$	0	0
$361$	−10.6474	−0.560390
$362$	16.1933	0.851100
$363$	3.75063	0.196857
$364$	0	0
$365$	0	0
$366$	5.84117	0.305323
$367$	−7.41358	−0.386986	−0.193493	−	0.981102i	$-0.561982\pi$
$367$	−7.41358	−0.386986	−0.193493	+	0.981102i	$0.561982\pi$
$368$	−6.78986	−0.353946
$369$	29.9095	1.55702
$370$	0	0
$371$	35.9952	1.86878
$372$	3.70171	0.191925
$373$	−29.6039	−1.53283	−0.766415	−	0.642345i	$-0.777961\pi$
$373$	−29.6039	−1.53283	−0.766415	+	0.642345i	$0.777961\pi$
$374$	−4.98792	−0.257919
$375$	0	0
$376$	3.97823	0.205162
$377$	0	0
$378$	8.38404	0.431229
$379$	−8.45042	−0.434069	−0.217034	−	0.976164i	$-0.569638\pi$
$379$	−8.45042	−0.434069	−0.217034	+	0.976164i	$0.569638\pi$
$380$	0	0
$381$	−1.24937	−0.0640073
$382$	−26.5676	−1.35932
$383$	−10.4644	−0.534707	−0.267353	−	0.963599i	$-0.586149\pi$
$383$	−10.4644	−0.534707	−0.267353	+	0.963599i	$0.586149\pi$
$384$	−0.445042	−0.0227109
$385$	0	0
$386$	−13.6582	−0.695183
$387$	−14.5308	−0.738642
$388$	−6.05429	−0.307360
$389$	29.2295	1.48200	0.740998	−	0.671507i	$-0.234353\pi$
$389$	29.2295	1.48200	0.740998	+	0.671507i	$0.234353\pi$
$390$	0	0
$391$	21.1159	1.06788
$392$	3.54288	0.178942
$393$	−1.05562	−0.0532491
$394$	14.0978	0.710239
$395$	0	0
$396$	−4.49396	−0.225830
$397$	−20.7439	−1.04111	−0.520554	−	0.853829i	$-0.674274\pi$
$397$	−20.7439	−1.04111	−0.520554	+	0.853829i	$0.674274\pi$
$398$	−21.1293	−1.05912
$399$	4.17629	0.209076
$400$	0	0
$401$	−24.2892	−1.21294	−0.606472	−	0.795105i	$-0.707416\pi$
$401$	−24.2892	−1.21294	−0.606472	+	0.795105i	$0.707416\pi$
$402$	−5.51573	−0.275100
$403$	0	0
$404$	−4.19806	−0.208861
$405$	0	0
$406$	13.1468	0.652462
$407$	−5.14483	−0.255020
$408$	1.38404	0.0685203
$409$	−36.7676	−1.81804	−0.909021	−	0.416751i	$-0.863169\pi$
$409$	−36.7676	−1.81804	−0.909021	+	0.416751i	$0.863169\pi$
$410$	0	0
$411$	−7.50125	−0.370010
$412$	−11.5211	−0.567604
$413$	−17.1642	−0.844596
$414$	19.0248	0.935016
$415$	0	0
$416$	0	0
$417$	−2.65950	−0.130236
$418$	−4.63533	−0.226722
$419$	−30.8418	−1.50672	−0.753359	−	0.657609i	$-0.771568\pi$
$419$	−30.8418	−1.50672	−0.753359	+	0.657609i	$0.771568\pi$
$420$	0	0
$421$	−31.7458	−1.54720	−0.773599	−	0.633676i	$-0.781545\pi$
$421$	−31.7458	−1.54720	−0.773599	+	0.633676i	$0.781545\pi$
$422$	−18.3720	−0.894333
$423$	−11.1468	−0.541974
$424$	11.0858	0.538372
$425$	0	0
$426$	3.34050	0.161848
$427$	−42.6165	−2.06236
$428$	4.43967	0.214599
$429$	0	0
$430$	0	0
$431$	25.5120	1.22887	0.614435	−	0.788967i	$-0.289384\pi$
$431$	25.5120	1.22887	0.614435	+	0.788967i	$0.289384\pi$
$432$	2.58211	0.124232
$433$	31.8189	1.52912	0.764560	−	0.644553i	$-0.222956\pi$
$433$	31.8189	1.52912	0.764560	+	0.644553i	$0.222956\pi$
$434$	−27.0073	−1.29639
$435$	0	0
$436$	10.5526	0.505376
$437$	19.6233	0.938707
$438$	5.46250	0.261008
$439$	26.9095	1.28432	0.642159	−	0.766571i	$-0.278039\pi$
$439$	26.9095	1.28432	0.642159	+	0.766571i	$0.278039\pi$
$440$	0	0
$441$	−9.92692	−0.472710
$442$	0	0
$443$	−12.7385	−0.605227	−0.302613	−	0.953113i	$-0.597859\pi$
$443$	−12.7385	−0.605227	−0.302613	+	0.953113i	$0.597859\pi$
$444$	1.42758	0.0677501
$445$	0	0
$446$	−14.2687	−0.675645
$447$	−6.63773	−0.313954
$448$	3.24698	0.153405
$449$	7.75063	0.365775	0.182887	−	0.983134i	$-0.441456\pi$
$449$	7.75063	0.365775	0.182887	+	0.983134i	$0.441456\pi$
$450$	0	0
$451$	−17.1207	−0.806181
$452$	−4.31767	−0.203086
$453$	−4.25475	−0.199906
$454$	−27.4959	−1.29045
$455$	0	0
$456$	1.28621	0.0602322
$457$	−6.94438	−0.324844	−0.162422	−	0.986721i	$-0.551931\pi$
$457$	−6.94438	−0.324844	−0.162422	+	0.986721i	$0.551931\pi$
$458$	19.6963	0.920349
$459$	−8.03013	−0.374814
$460$	0	0
$461$	−12.6377	−0.588598	−0.294299	−	0.955713i	$-0.595086\pi$
$461$	−12.6377	−0.588598	−0.294299	+	0.955713i	$0.595086\pi$
$462$	−2.31767	−0.107828
$463$	5.62325	0.261335	0.130667	−	0.991426i	$-0.458288\pi$
$463$	5.62325	0.261335	0.130667	+	0.991426i	$0.458288\pi$
$464$	4.04892	0.187966
$465$	0	0
$466$	23.5362	1.09029
$467$	35.2054	1.62911	0.814555	−	0.580087i	$-0.196981\pi$
$467$	35.2054	1.62911	0.814555	+	0.580087i	$0.196981\pi$
$468$	0	0
$469$	40.2422	1.85821
$470$	0	0
$471$	−6.12200	−0.282087
$472$	−5.28621	−0.243317
$473$	8.31767	0.382447
$474$	5.03146	0.231103
$475$	0	0
$476$	−10.0978	−0.462833
$477$	−31.0616	−1.42221
$478$	−14.4155	−0.659350
$479$	7.16421	0.327341	0.163671	−	0.986515i	$-0.447667\pi$
$479$	7.16421	0.327341	0.163671	+	0.986515i	$0.447667\pi$
$480$	0	0
$481$	0	0
$482$	15.6692	0.713712
$483$	9.81163	0.446444
$484$	−8.42758	−0.383072
$485$	0	0
$486$	−10.9758	−0.497874
$487$	−28.7006	−1.30055	−0.650275	−	0.759699i	$-0.725346\pi$
$487$	−28.7006	−1.30055	−0.650275	+	0.759699i	$0.725346\pi$
$488$	−13.1250	−0.594140
$489$	−6.02177	−0.272314
$490$	0	0
$491$	−42.0689	−1.89854	−0.949271	−	0.314459i	$-0.898177\pi$
$491$	−42.0689	−1.89854	−0.949271	+	0.314459i	$0.898177\pi$
$492$	4.75063	0.214175
$493$	−12.5918	−0.567106
$494$	0	0
$495$	0	0
$496$	−8.31767	−0.373474
$497$	−24.3720	−1.09323
$498$	4.63773	0.207822
$499$	33.2379	1.48793	0.743966	−	0.668218i	$-0.232942\pi$
$499$	33.2379	1.48793	0.743966	+	0.668218i	$0.232942\pi$
$500$	0	0
$501$	−6.75063	−0.301596
$502$	−15.1535	−0.676332
$503$	37.7797	1.68451	0.842257	−	0.539077i	$-0.181227\pi$
$503$	37.7797	1.68451	0.842257	+	0.539077i	$0.181227\pi$
$504$	−9.09783	−0.405250
$505$	0	0
$506$	−10.8901	−0.484123
$507$	0	0
$508$	2.80731	0.124554
$509$	2.26875	0.100561	0.0502803	−	0.998735i	$-0.483989\pi$
$509$	2.26875	0.100561	0.0502803	+	0.998735i	$0.483989\pi$
$510$	0	0
$511$	−39.8538	−1.76303
$512$	1.00000	0.0441942
$513$	−7.46250	−0.329477
$514$	−18.8552	−0.831666
$515$	0	0
$516$	−2.30798	−0.101603
$517$	6.38059	0.280618
$518$	−10.4155	−0.457631
$519$	1.90217	0.0834958
$520$	0	0
$521$	−22.7899	−0.998442	−0.499221	−	0.866475i	$-0.666380\pi$
$521$	−22.7899	−0.998442	−0.499221	+	0.866475i	$0.666380\pi$
$522$	−11.3448	−0.496549
$523$	−6.61356	−0.289191	−0.144595	−	0.989491i	$-0.546188\pi$
$523$	−6.61356	−0.289191	−0.144595	+	0.989491i	$0.546188\pi$
$524$	2.37196	0.103620
$525$	0	0
$526$	2.18060	0.0950788
$527$	25.8672	1.12680
$528$	−0.713792	−0.0310638
$529$	23.1021	1.00444
$530$	0	0
$531$	14.8116	0.642770
$532$	−9.38404	−0.406850
$533$	0	0
$534$	5.06531	0.219197
$535$	0	0
$536$	12.3937	0.535328
$537$	−6.95300	−0.300044
$538$	9.73125	0.419544
$539$	5.68233	0.244755
$540$	0	0
$541$	24.2553	1.04282	0.521409	−	0.853307i	$-0.325407\pi$
$541$	24.2553	1.04282	0.521409	+	0.853307i	$0.325407\pi$
$542$	4.05429	0.174147
$543$	−7.20669	−0.309268
$544$	−3.10992	−0.133337
$545$	0	0
$546$	0	0
$547$	−11.8086	−0.504901	−0.252451	−	0.967610i	$-0.581237\pi$
$547$	−11.8086	−0.504901	−0.252451	+	0.967610i	$0.581237\pi$
$548$	16.8552	0.720017
$549$	36.7754	1.56954
$550$	0	0
$551$	−11.7017	−0.498510
$552$	3.02177	0.128615
$553$	−36.7090	−1.56103
$554$	−30.0301	−1.27586
$555$	0	0
$556$	5.97584	0.253432
$557$	15.0180	0.636335	0.318168	−	0.948034i	$-0.396933\pi$
$557$	15.0180	0.636335	0.318168	+	0.948034i	$0.396933\pi$
$558$	23.3056	0.986604
$559$	0	0
$560$	0	0
$561$	2.21983	0.0937214
$562$	22.2640	0.939149
$563$	−10.4198	−0.439143	−0.219571	−	0.975596i	$-0.570466\pi$
$563$	−10.4198	−0.439143	−0.219571	+	0.975596i	$0.570466\pi$
$564$	−1.77048	−0.0745506
$565$	0	0
$566$	8.88471	0.373452
$567$	23.5623	0.989522
$568$	−7.50604	−0.314946
$569$	12.3461	0.517577	0.258789	−	0.965934i	$-0.416677\pi$
$569$	12.3461	0.517577	0.258789	+	0.965934i	$0.416677\pi$
$570$	0	0
$571$	−2.21983	−0.0928971	−0.0464486	−	0.998921i	$-0.514790\pi$
$571$	−2.21983	−0.0928971	−0.0464486	+	0.998921i	$0.514790\pi$
$572$	0	0
$573$	11.8237	0.493942
$574$	−34.6601	−1.44668
$575$	0	0
$576$	−2.80194	−0.116747
$577$	21.3405	0.888417	0.444208	−	0.895924i	$-0.353485\pi$
$577$	21.3405	0.888417	0.444208	+	0.895924i	$0.353485\pi$
$578$	−7.32842	−0.304822
$579$	6.07846	0.252612
$580$	0	0
$581$	−33.8364	−1.40377
$582$	2.69441	0.111687
$583$	17.7802	0.736379
$584$	−12.2741	−0.507907
$585$	0	0
$586$	−21.5362	−0.889651
$587$	10.1371	0.418401	0.209201	−	0.977873i	$-0.432914\pi$
$587$	10.1371	0.418401	0.209201	+	0.977873i	$0.432914\pi$
$588$	−1.57673	−0.0650232
$589$	24.0388	0.990500
$590$	0	0
$591$	−6.27413	−0.258083
$592$	−3.20775	−0.131838
$593$	−17.6668	−0.725488	−0.362744	−	0.931889i	$-0.618160\pi$
$593$	−17.6668	−0.725488	−0.362744	+	0.931889i	$0.618160\pi$
$594$	4.14138	0.169923
$595$	0	0
$596$	14.9148	0.610936
$597$	9.40342	0.384856
$598$	0	0
$599$	9.14005	0.373452	0.186726	−	0.982412i	$-0.440212\pi$
$599$	9.14005	0.373452	0.186726	+	0.982412i	$0.440212\pi$
$600$	0	0
$601$	37.0116	1.50973	0.754867	−	0.655877i	$-0.227701\pi$
$601$	37.0116	1.50973	0.754867	+	0.655877i	$0.227701\pi$
$602$	16.8388	0.686297
$603$	−34.7265	−1.41417
$604$	9.56033	0.389005
$605$	0	0
$606$	1.86831	0.0758950
$607$	−10.9420	−0.444121	−0.222061	−	0.975033i	$-0.571278\pi$
$607$	−10.9420	−0.444121	−0.222061	+	0.975033i	$0.571278\pi$
$608$	−2.89008	−0.117208
$609$	−5.85086	−0.237089
$610$	0	0
$611$	0	0
$612$	8.71379	0.352234
$613$	−10.9336	−0.441605	−0.220802	−	0.975319i	$-0.570868\pi$
$613$	−10.9336	−0.441605	−0.220802	+	0.975319i	$0.570868\pi$
$614$	10.3230	0.416604
$615$	0	0
$616$	5.20775	0.209826
$617$	−6.33704	−0.255120	−0.127560	−	0.991831i	$-0.540715\pi$
$617$	−6.33704	−0.255120	−0.127560	+	0.991831i	$0.540715\pi$
$618$	5.12737	0.206253
$619$	12.8659	0.517125	0.258563	−	0.965995i	$-0.416751\pi$
$619$	12.8659	0.517125	0.258563	+	0.965995i	$0.416751\pi$
$620$	0	0
$621$	−17.5321	−0.703540
$622$	7.50125	0.300773
$623$	−36.9560	−1.48061
$624$	0	0
$625$	0	0
$626$	−5.60388	−0.223976
$627$	2.06292	0.0823850
$628$	13.7560	0.548924
$629$	9.97584	0.397763
$630$	0	0
$631$	−47.8491	−1.90484	−0.952420	−	0.304788i	$-0.901415\pi$
$631$	−47.8491	−1.90484	−0.952420	+	0.304788i	$0.901415\pi$
$632$	−11.3056	−0.449712
$633$	8.17629	0.324978
$634$	11.7802	0.467850
$635$	0	0
$636$	−4.93362	−0.195631
$637$	0	0
$638$	6.49396	0.257098
$639$	21.0315	0.831992
$640$	0	0
$641$	45.7168	1.80570	0.902852	−	0.429951i	$-0.141469\pi$
$641$	45.7168	1.80570	0.902852	+	0.429951i	$0.141469\pi$
$642$	−1.97584	−0.0779801
$643$	33.1836	1.30863	0.654316	−	0.756221i	$-0.272956\pi$
$643$	33.1836	1.30863	0.654316	+	0.756221i	$0.272956\pi$
$644$	−22.0465	−0.868755
$645$	0	0
$646$	8.98792	0.353625
$647$	9.53617	0.374906	0.187453	−	0.982274i	$-0.439977\pi$
$647$	9.53617	0.374906	0.187453	+	0.982274i	$0.439977\pi$
$648$	7.25667	0.285069
$649$	−8.47842	−0.332807
$650$	0	0
$651$	12.0194	0.471077
$652$	13.5308	0.529907
$653$	40.5918	1.58848	0.794240	−	0.607604i	$-0.207869\pi$
$653$	40.5918	1.58848	0.794240	+	0.607604i	$0.207869\pi$
$654$	−4.69633	−0.183641
$655$	0	0
$656$	−10.6746	−0.416772
$657$	34.3913	1.34173
$658$	12.9172	0.503566
$659$	−0.537500	−0.0209380	−0.0104690	−	0.999945i	$-0.503332\pi$
$659$	−0.537500	−0.0209380	−0.0104690	+	0.999945i	$0.503332\pi$
$660$	0	0
$661$	23.9928	0.933213	0.466606	−	0.884465i	$-0.345477\pi$
$661$	23.9928	0.933213	0.466606	+	0.884465i	$0.345477\pi$
$662$	−6.46980	−0.251456
$663$	0	0
$664$	−10.4209	−0.404409
$665$	0	0
$666$	8.98792	0.348275
$667$	−27.4916	−1.06448
$668$	15.1685	0.586888
$669$	6.35019	0.245513
$670$	0	0
$671$	−21.0508	−0.812659
$672$	−1.44504	−0.0557437
$673$	−7.26683	−0.280116	−0.140058	−	0.990143i	$-0.544729\pi$
$673$	−7.26683	−0.280116	−0.140058	+	0.990143i	$0.544729\pi$
$674$	5.35988	0.206455
$675$	0	0
$676$	0	0
$677$	−5.11470	−0.196574	−0.0982870	−	0.995158i	$-0.531336\pi$
$677$	−5.11470	−0.196574	−0.0982870	+	0.995158i	$0.531336\pi$
$678$	1.92154	0.0737964
$679$	−19.6582	−0.754411
$680$	0	0
$681$	12.2368	0.468916
$682$	−13.3405	−0.510834
$683$	23.0532	0.882107	0.441054	−	0.897481i	$-0.354605\pi$
$683$	23.0532	0.882107	0.441054	+	0.897481i	$0.354605\pi$
$684$	8.09783	0.309628
$685$	0	0
$686$	−11.2252	−0.428580
$687$	−8.76569	−0.334432
$688$	5.18598	0.197714
$689$	0	0
$690$	0	0
$691$	31.1051	1.18329	0.591647	−	0.806197i	$-0.298478\pi$
$691$	31.1051	1.18329	0.591647	+	0.806197i	$0.298478\pi$
$692$	−4.27413	−0.162478
$693$	−14.5918	−0.554296
$694$	−3.56571	−0.135353
$695$	0	0
$696$	−1.80194	−0.0683023
$697$	33.1970	1.25743
$698$	−7.97584	−0.301890
$699$	−10.4746	−0.396185
$700$	0	0
$701$	7.07069	0.267056	0.133528	−	0.991045i	$-0.457369\pi$
$701$	7.07069	0.267056	0.133528	+	0.991045i	$0.457369\pi$
$702$	0	0
$703$	9.27067	0.349650
$704$	1.60388	0.0604483
$705$	0	0
$706$	1.60388	0.0603626
$707$	−13.6310	−0.512647
$708$	2.35258	0.0884155
$709$	8.65519	0.325052	0.162526	−	0.986704i	$-0.448036\pi$
$709$	8.65519	0.325052	0.162526	+	0.986704i	$0.448036\pi$
$710$	0	0
$711$	31.6775	1.18800
$712$	−11.3817	−0.426545
$713$	56.4758	2.11503
$714$	4.49396	0.168182
$715$	0	0
$716$	15.6233	0.583868
$717$	6.41550	0.239591
$718$	−8.60255	−0.321044
$719$	−8.48055	−0.316271	−0.158136	−	0.987417i	$-0.550548\pi$
$719$	−8.48055	−0.316271	−0.158136	+	0.987417i	$0.550548\pi$
$720$	0	0
$721$	−37.4088	−1.39318
$722$	−10.6474	−0.396256
$723$	−6.97344	−0.259345
$724$	16.1933	0.601818
$725$	0	0
$726$	3.75063	0.139199
$727$	−31.2683	−1.15968	−0.579838	−	0.814732i	$-0.696884\pi$
$727$	−31.2683	−1.15968	−0.579838	+	0.814732i	$0.696884\pi$
$728$	0	0
$729$	−16.8853	−0.625381
$730$	0	0
$731$	−16.1280	−0.596514
$732$	5.84117	0.215896
$733$	−6.69441	−0.247264	−0.123632	−	0.992328i	$-0.539454\pi$
$733$	−6.69441	−0.247264	−0.123632	+	0.992328i	$0.539454\pi$
$734$	−7.41358	−0.273640
$735$	0	0
$736$	−6.78986	−0.250277
$737$	19.8780	0.732216
$738$	29.9095	1.10098
$739$	−5.29696	−0.194852	−0.0974259	−	0.995243i	$-0.531061\pi$
$739$	−5.29696	−0.194852	−0.0974259	+	0.995243i	$0.531061\pi$
$740$	0	0
$741$	0	0
$742$	35.9952	1.32143
$743$	13.0755	0.479693	0.239846	−	0.970811i	$-0.422903\pi$
$743$	13.0755	0.479693	0.239846	+	0.970811i	$0.422903\pi$
$744$	3.70171	0.135711
$745$	0	0
$746$	−29.6039	−1.08387
$747$	29.1987	1.06832
$748$	−4.98792	−0.182376
$749$	14.4155	0.526731
$750$	0	0
$751$	−26.6461	−0.972330	−0.486165	−	0.873867i	$-0.661605\pi$
$751$	−26.6461	−0.972330	−0.486165	+	0.873867i	$0.661605\pi$
$752$	3.97823	0.145071
$753$	6.74392	0.245762
$754$	0	0
$755$	0	0
$756$	8.38404	0.304925
$757$	−54.3236	−1.97443	−0.987213	−	0.159406i	$-0.949042\pi$
$757$	−54.3236	−1.97443	−0.987213	+	0.159406i	$0.949042\pi$
$758$	−8.45042	−0.306933
$759$	4.84654	0.175918
$760$	0	0
$761$	−8.55363	−0.310069	−0.155034	−	0.987909i	$-0.549549\pi$
$761$	−8.55363	−0.310069	−0.155034	+	0.987909i	$0.549549\pi$
$762$	−1.24937	−0.0452600
$763$	34.2640	1.24044
$764$	−26.5676	−0.961183
$765$	0	0
$766$	−10.4644	−0.378095
$767$	0	0
$768$	−0.445042	−0.0160591
$769$	−20.6504	−0.744672	−0.372336	−	0.928098i	$-0.621443\pi$
$769$	−20.6504	−0.744672	−0.372336	+	0.928098i	$0.621443\pi$
$770$	0	0
$771$	8.39134	0.302207
$772$	−13.6582	−0.491568
$773$	5.27545	0.189745	0.0948725	−	0.995489i	$-0.469756\pi$
$773$	5.27545	0.189745	0.0948725	+	0.995489i	$0.469756\pi$
$774$	−14.5308	−0.522299
$775$	0	0
$776$	−6.05429	−0.217336
$777$	4.63533	0.166292
$778$	29.2295	1.04793
$779$	30.8504	1.10533
$780$	0	0
$781$	−12.0388	−0.430781
$782$	21.1159	0.755102
$783$	10.4547	0.373622
$784$	3.54288	0.126531
$785$	0	0
$786$	−1.05562	−0.0376528
$787$	39.7077	1.41543	0.707713	−	0.706500i	$-0.249727\pi$
$787$	39.7077	1.41543	0.707713	+	0.706500i	$0.249727\pi$
$788$	14.0978	0.502215
$789$	−0.970460	−0.0345493
$790$	0	0
$791$	−14.0194	−0.498472
$792$	−4.49396	−0.159686
$793$	0	0
$794$	−20.7439	−0.736174
$795$	0	0
$796$	−21.1293	−0.748908
$797$	−36.5676	−1.29529	−0.647646	−	0.761941i	$-0.724246\pi$
$797$	−36.5676	−1.29529	−0.647646	+	0.761941i	$0.724246\pi$
$798$	4.17629	0.147839
$799$	−12.3720	−0.437689
$800$	0	0
$801$	31.8907	1.12680
$802$	−24.2892	−0.857681
$803$	−19.6862	−0.694710
$804$	−5.51573	−0.194525
$805$	0	0
$806$	0	0
$807$	−4.33081	−0.152452
$808$	−4.19806	−0.147687
$809$	−21.9694	−0.772403	−0.386201	−	0.922414i	$-0.626213\pi$
$809$	−21.9694	−0.772403	−0.386201	+	0.922414i	$0.626213\pi$
$810$	0	0
$811$	31.3250	1.09997	0.549984	−	0.835175i	$-0.314634\pi$
$811$	31.3250	1.09997	0.549984	+	0.835175i	$0.314634\pi$
$812$	13.1468	0.461361
$813$	−1.80433	−0.0632806
$814$	−5.14483	−0.180326
$815$	0	0
$816$	1.38404	0.0484512
$817$	−14.9879	−0.524361
$818$	−36.7676	−1.28555
$819$	0	0
$820$	0	0
$821$	−13.9734	−0.487677	−0.243838	−	0.969816i	$-0.578407\pi$
$821$	−13.9734	−0.487677	−0.243838	+	0.969816i	$0.578407\pi$
$822$	−7.50125	−0.261636
$823$	−32.0871	−1.11849	−0.559243	−	0.829004i	$-0.688908\pi$
$823$	−32.0871	−1.11849	−0.559243	+	0.829004i	$0.688908\pi$
$824$	−11.5211	−0.401357
$825$	0	0
$826$	−17.1642	−0.597219
$827$	−1.82430	−0.0634371	−0.0317185	−	0.999497i	$-0.510098\pi$
$827$	−1.82430	−0.0634371	−0.0317185	+	0.999497i	$0.510098\pi$
$828$	19.0248	0.661156
$829$	−32.1360	−1.11613	−0.558065	−	0.829797i	$-0.688456\pi$
$829$	−32.1360	−1.11613	−0.558065	+	0.829797i	$0.688456\pi$
$830$	0	0
$831$	13.3647	0.463615
$832$	0	0
$833$	−11.0180	−0.381753
$834$	−2.65950	−0.0920909
$835$	0	0
$836$	−4.63533	−0.160316
$837$	−21.4771	−0.742357
$838$	−30.8418	−1.06541
$839$	−8.30904	−0.286860	−0.143430	−	0.989660i	$-0.545813\pi$
$839$	−8.30904	−0.286860	−0.143430	+	0.989660i	$0.545813\pi$
$840$	0	0
$841$	−12.6063	−0.434699
$842$	−31.7458	−1.09403
$843$	−9.90840	−0.341263
$844$	−18.3720	−0.632389
$845$	0	0
$846$	−11.1468	−0.383233
$847$	−27.3642	−0.940245
$848$	11.0858	0.380686
$849$	−3.95407	−0.135703
$850$	0	0
$851$	21.7802	0.746615
$852$	3.34050	0.114444
$853$	−31.7995	−1.08880	−0.544398	−	0.838827i	$-0.683242\pi$
$853$	−31.7995	−1.08880	−0.544398	+	0.838827i	$0.683242\pi$
$854$	−42.6165	−1.45831
$855$	0	0
$856$	4.43967	0.151745
$857$	9.32975	0.318698	0.159349	−	0.987222i	$-0.449060\pi$
$857$	9.32975	0.318698	0.159349	+	0.987222i	$0.449060\pi$
$858$	0	0
$859$	−58.3913	−1.99229	−0.996143	−	0.0877403i	$-0.972035\pi$
$859$	−58.3913	−1.99229	−0.996143	+	0.0877403i	$0.972035\pi$
$860$	0	0
$861$	15.4252	0.525689
$862$	25.5120	0.868942
$863$	−8.79284	−0.299312	−0.149656	−	0.988738i	$-0.547817\pi$
$863$	−8.79284	−0.299312	−0.149656	+	0.988738i	$0.547817\pi$
$864$	2.58211	0.0878450
$865$	0	0
$866$	31.8189	1.08125
$867$	3.26145	0.110765
$868$	−27.0073	−0.916687
$869$	−18.1328	−0.615111
$870$	0	0
$871$	0	0
$872$	10.5526	0.357355
$873$	16.9638	0.574136
$874$	19.6233	0.663766
$875$	0	0
$876$	5.46250	0.184561
$877$	33.6668	1.13685	0.568423	−	0.822736i	$-0.307554\pi$
$877$	33.6668	1.13685	0.568423	+	0.822736i	$0.307554\pi$
$878$	26.9095	0.908150
$879$	9.58450	0.323277
$880$	0	0
$881$	24.8068	0.835764	0.417882	−	0.908501i	$-0.362773\pi$
$881$	24.8068	0.835764	0.417882	+	0.908501i	$0.362773\pi$
$882$	−9.92692	−0.334257
$883$	−15.5120	−0.522021	−0.261010	−	0.965336i	$-0.584056\pi$
$883$	−15.5120	−0.522021	−0.261010	+	0.965336i	$0.584056\pi$
$884$	0	0
$885$	0	0
$886$	−12.7385	−0.427960
$887$	12.9487	0.434774	0.217387	−	0.976085i	$-0.430247\pi$
$887$	12.9487	0.434774	0.217387	+	0.976085i	$0.430247\pi$
$888$	1.42758	0.0479066
$889$	9.11529	0.305717
$890$	0	0
$891$	11.6388	0.389914
$892$	−14.2687	−0.477753
$893$	−11.4974	−0.384746
$894$	−6.63773	−0.221999
$895$	0	0
$896$	3.24698	0.108474
$897$	0	0
$898$	7.75063	0.258642
$899$	−33.6775	−1.12321
$900$	0	0
$901$	−34.4758	−1.14855
$902$	−17.1207	−0.570056
$903$	−7.49396	−0.249383
$904$	−4.31767	−0.143603
$905$	0	0
$906$	−4.25475	−0.141355
$907$	26.4983	0.879861	0.439930	−	0.898032i	$-0.355003\pi$
$907$	26.4983	0.879861	0.439930	+	0.898032i	$0.355003\pi$
$908$	−27.4959	−0.912483
$909$	11.7627	0.390144
$910$	0	0
$911$	−15.5797	−0.516179	−0.258089	−	0.966121i	$-0.583093\pi$
$911$	−15.5797	−0.516179	−0.258089	+	0.966121i	$0.583093\pi$
$912$	1.28621	0.0425906
$913$	−16.7138	−0.553146
$914$	−6.94438	−0.229700
$915$	0	0
$916$	19.6963	0.650785
$917$	7.70171	0.254333
$918$	−8.03013	−0.265034
$919$	−44.5870	−1.47079	−0.735395	−	0.677639i	$-0.763003\pi$
$919$	−44.5870	−1.47079	−0.735395	+	0.677639i	$0.763003\pi$
$920$	0	0
$921$	−4.59419	−0.151384
$922$	−12.6377	−0.416201
$923$	0	0
$924$	−2.31767	−0.0762457
$925$	0	0
$926$	5.62325	0.184792
$927$	32.2814	1.06026
$928$	4.04892	0.132912
$929$	7.09890	0.232907	0.116454	−	0.993196i	$-0.462847\pi$
$929$	7.09890	0.232907	0.116454	+	0.993196i	$0.462847\pi$
$930$	0	0
$931$	−10.2392	−0.335577
$932$	23.5362	0.770953
$933$	−3.33837	−0.109293
$934$	35.2054	1.15195
$935$	0	0
$936$	0	0
$937$	11.3297	0.370127	0.185063	−	0.982727i	$-0.440751\pi$
$937$	11.3297	0.370127	0.185063	+	0.982727i	$0.440751\pi$
$938$	40.2422	1.31395
$939$	2.49396	0.0813873
$940$	0	0
$941$	49.8689	1.62568	0.812840	−	0.582487i	$-0.197920\pi$
$941$	49.8689	1.62568	0.812840	+	0.582487i	$0.197920\pi$
$942$	−6.12200	−0.199465
$943$	72.4787	2.36023
$944$	−5.28621	−0.172051
$945$	0	0
$946$	8.31767	0.270431
$947$	22.2457	0.722887	0.361443	−	0.932394i	$-0.382284\pi$
$947$	22.2457	0.722887	0.361443	+	0.932394i	$0.382284\pi$
$948$	5.03146	0.163414
$949$	0	0
$950$	0	0
$951$	−5.24267	−0.170005
$952$	−10.0978	−0.327273
$953$	−35.3900	−1.14639	−0.573197	−	0.819417i	$-0.694297\pi$
$953$	−35.3900	−1.14639	−0.573197	+	0.819417i	$0.694297\pi$
$954$	−31.0616	−1.00566
$955$	0	0
$956$	−14.4155	−0.466231
$957$	−2.89008	−0.0934231
$958$	7.16421	0.231465
$959$	54.7284	1.76727
$960$	0	0
$961$	38.1836	1.23173
$962$	0	0
$963$	−12.4397	−0.400863
$964$	15.6692	0.504671
$965$	0	0
$966$	9.81163	0.315684
$967$	51.0723	1.64238	0.821188	−	0.570658i	$-0.193312\pi$
$967$	51.0723	1.64238	0.821188	+	0.570658i	$0.193312\pi$
$968$	−8.42758	−0.270873
$969$	−4.00000	−0.128499
$970$	0	0
$971$	37.8840	1.21575	0.607877	−	0.794031i	$-0.292021\pi$
$971$	37.8840	1.21575	0.607877	+	0.794031i	$0.292021\pi$
$972$	−10.9758	−0.352050
$973$	19.4034	0.622045
$974$	−28.7006	−0.919628
$975$	0	0
$976$	−13.1250	−0.420120
$977$	31.3685	1.00357	0.501784	−	0.864993i	$-0.332677\pi$
$977$	31.3685	1.00357	0.501784	+	0.864993i	$0.332677\pi$
$978$	−6.02177	−0.192555
$979$	−18.2547	−0.583424
$980$	0	0
$981$	−29.5676	−0.944022
$982$	−42.0689	−1.34247
$983$	39.3653	1.25556	0.627778	−	0.778392i	$-0.283964\pi$
$983$	39.3653	1.25556	0.627778	+	0.778392i	$0.283964\pi$
$984$	4.75063	0.151444
$985$	0	0
$986$	−12.5918	−0.401004
$987$	−5.74871	−0.182983
$988$	0	0
$989$	−35.2121	−1.11968
$990$	0	0
$991$	43.2922	1.37522	0.687611	−	0.726080i	$-0.258660\pi$
$991$	43.2922	1.37522	0.687611	+	0.726080i	$0.258660\pi$
$992$	−8.31767	−0.264086
$993$	2.87933	0.0913728
$994$	−24.3720	−0.773032
$995$	0	0
$996$	4.63773	0.146952
$997$	4.64609	0.147143	0.0735715	−	0.997290i	$-0.476560\pi$
$997$	4.64609	0.147143	0.0735715	+	0.997290i	$0.476560\pi$
$998$	33.2379	1.05213
$999$	−8.28275	−0.262055

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.bz.1.2		3
5.4	even	2		1690.2.a.q.1.2	✓	3
13.12	even	2		8450.2.a.bo.1.2		3
65.4	even	6		1690.2.e.o.991.2		6
65.9	even	6		1690.2.e.q.991.2		6
65.19	odd	12		1690.2.l.l.361.5		12
65.24	odd	12		1690.2.l.l.1161.5		12
65.29	even	6		1690.2.e.q.191.2		6
65.34	odd	4		1690.2.d.j.1351.2		6
65.44	odd	4		1690.2.d.j.1351.5		6
65.49	even	6		1690.2.e.o.191.2		6
65.54	odd	12		1690.2.l.l.1161.2		12
65.59	odd	12		1690.2.l.l.361.2		12
65.64	even	2		1690.2.a.s.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.a.q.1.2	✓	3	5.4	even	2
1690.2.a.s.1.2	yes	3	65.64	even	2
1690.2.d.j.1351.2		6	65.34	odd	4
1690.2.d.j.1351.5		6	65.44	odd	4
1690.2.e.o.191.2		6	65.49	even	6
1690.2.e.o.991.2		6	65.4	even	6
1690.2.e.q.191.2		6	65.29	even	6
1690.2.e.q.991.2		6	65.9	even	6
1690.2.l.l.361.2		12	65.59	odd	12
1690.2.l.l.361.5		12	65.19	odd	12
1690.2.l.l.1161.2		12	65.54	odd	12
1690.2.l.l.1161.5		12	65.24	odd	12
8450.2.a.bo.1.2		3	13.12	even	2
8450.2.a.bz.1.2		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} -0.445042 q^{6} +3.24698 q^{7} +1.00000 q^{8} -2.80194 q^{9} +1.60388 q^{11} -0.445042 q^{12} +3.24698 q^{14} +1.00000 q^{16} -3.10992 q^{17} -2.80194 q^{18} -2.89008 q^{19} -1.44504 q^{21} +1.60388 q^{22} -6.78986 q^{23} -0.445042 q^{24} +2.58211 q^{27} +3.24698 q^{28} +4.04892 q^{29} -8.31767 q^{31} +1.00000 q^{32} -0.713792 q^{33} -3.10992 q^{34} -2.80194 q^{36} -3.20775 q^{37} -2.89008 q^{38} -10.6746 q^{41} -1.44504 q^{42} +5.18598 q^{43} +1.60388 q^{44} -6.78986 q^{46} +3.97823 q^{47} -0.445042 q^{48} +3.54288 q^{49} +1.38404 q^{51} +11.0858 q^{53} +2.58211 q^{54} +3.24698 q^{56} +1.28621 q^{57} +4.04892 q^{58} -5.28621 q^{59} -13.1250 q^{61} -8.31767 q^{62} -9.09783 q^{63} +1.00000 q^{64} -0.713792 q^{66} +12.3937 q^{67} -3.10992 q^{68} +3.02177 q^{69} -7.50604 q^{71} -2.80194 q^{72} -12.2741 q^{73} -3.20775 q^{74} -2.89008 q^{76} +5.20775 q^{77} -11.3056 q^{79} +7.25667 q^{81} -10.6746 q^{82} -10.4209 q^{83} -1.44504 q^{84} +5.18598 q^{86} -1.80194 q^{87} +1.60388 q^{88} -11.3817 q^{89} -6.78986 q^{92} +3.70171 q^{93} +3.97823 q^{94} -0.445042 q^{96} -6.05429 q^{97} +3.54288 q^{98} -4.49396 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} + 5 q^{7} + 3 q^{8} - 4 q^{9} - 4 q^{11} - q^{12} + 5 q^{14} + 3 q^{16} - 10 q^{17} - 4 q^{18} - 8 q^{19} - 4 q^{21} - 4 q^{22} + 3 q^{23} - q^{24} + 2 q^{27}+ \cdots - 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^6 + 5 * q^7 + 3 * q^8 - 4 * q^9 - 4 * q^11 - q^12 + 5 * q^14 + 3 * q^16 - 10 * q^17 - 4 * q^18 - 8 * q^19 - 4 * q^21 - 4 * q^22 + 3 * q^23 - q^24 + 2 * q^27 + 5 * q^28 + 3 * q^29 - 8 * q^31 + 3 * q^32 + 6 * q^33 - 10 * q^34 - 4 * q^36 + 8 * q^37 - 8 * q^38 - 11 * q^41 - 4 * q^42 + q^43 - 4 * q^44 + 3 * q^46 + 15 * q^47 - q^48 - 8 * q^49 - 6 * q^51 - 4 * q^53 + 2 * q^54 + 5 * q^56 + 12 * q^57 + 3 * q^58 - 24 * q^59 - 15 * q^61 - 8 * q^62 - 9 * q^63 + 3 * q^64 + 6 * q^66 + 5 * q^67 - 10 * q^68 + 6 * q^69 - 32 * q^71 - 4 * q^72 - 26 * q^73 + 8 * q^74 - 8 * q^76 - 2 * q^77 + 2 * q^79 - 5 * q^81 - 11 * q^82 + 7 * q^83 - 4 * q^84 + q^86 - q^87 - 4 * q^88 + 17 * q^89 + 3 * q^92 - 16 * q^93 + 15 * q^94 - q^96 - 6 * q^97 - 8 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more