LMFDB - Embedded newform 6003.2.a.t.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6003,2,Mod(1,6003)] 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("6003.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6003, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.a (trivial)

Newform invariants

comment:select newform

sage:traces = [22,-3,0,17,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	6003.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.24483 q^{2} +3.03928 q^{4} +2.70888 q^{5} -1.81542 q^{7} -2.33301 q^{8} -6.08099 q^{10} +1.28860 q^{11} +4.53826 q^{13} +4.07531 q^{14} -0.841340 q^{16} -4.48595 q^{17} +2.67238 q^{19} +8.23305 q^{20} -2.89270 q^{22} +1.00000 q^{23} +2.33804 q^{25} -10.1876 q^{26} -5.51756 q^{28} +1.00000 q^{29} -5.79860 q^{31} +6.55469 q^{32} +10.0702 q^{34} -4.91775 q^{35} -5.91943 q^{37} -5.99906 q^{38} -6.31985 q^{40} -7.82902 q^{41} -5.28362 q^{43} +3.91642 q^{44} -2.24483 q^{46} +2.52885 q^{47} -3.70427 q^{49} -5.24851 q^{50} +13.7930 q^{52} +7.14230 q^{53} +3.49067 q^{55} +4.23538 q^{56} -2.24483 q^{58} -7.87478 q^{59} +0.0192849 q^{61} +13.0169 q^{62} -13.0315 q^{64} +12.2936 q^{65} -15.5222 q^{67} -13.6340 q^{68} +11.0395 q^{70} +8.18804 q^{71} -16.8537 q^{73} +13.2881 q^{74} +8.12212 q^{76} -2.33935 q^{77} +2.32730 q^{79} -2.27909 q^{80} +17.5749 q^{82} +8.65460 q^{83} -12.1519 q^{85} +11.8608 q^{86} -3.00632 q^{88} +17.6714 q^{89} -8.23883 q^{91} +3.03928 q^{92} -5.67686 q^{94} +7.23917 q^{95} -7.50809 q^{97} +8.31546 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34}+ \cdots - 28 q^{98}+O(q^{100}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

Coefficient data

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

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

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

$2$	−2.24483	−1.58734	−0.793669	−	0.608350i	$-0.791832\pi$
$2$	−2.24483	−1.58734	−0.793669	+	0.608350i	$0.791832\pi$
$3$	0	0
$3$	0	0
$4$	3.03928	1.51964
$5$	2.70888	1.21145	0.605724	−	0.795675i	$-0.292883\pi$
$5$	2.70888	1.21145	0.605724	+	0.795675i	$0.292883\pi$
$6$	0	0
$7$	−1.81542	−0.686163	−0.343081	−	0.939306i	$-0.611471\pi$
$7$	−1.81542	−0.686163	−0.343081	+	0.939306i	$0.611471\pi$
$8$	−2.33301	−0.824843
$9$	0	0
$10$	−6.08099	−1.92298
$11$	1.28860	0.388528	0.194264	−	0.980949i	$-0.437768\pi$
$11$	1.28860	0.388528	0.194264	+	0.980949i	$0.437768\pi$
$12$	0	0
$13$	4.53826	1.25869	0.629343	−	0.777127i	$-0.283324\pi$
$13$	4.53826	1.25869	0.629343	+	0.777127i	$0.283324\pi$
$14$	4.07531	1.08917
$15$	0	0
$16$	−0.841340	−0.210335
$17$	−4.48595	−1.08800	−0.544001	−	0.839085i	$-0.683091\pi$
$17$	−4.48595	−1.08800	−0.544001	+	0.839085i	$0.683091\pi$
$18$	0	0
$19$	2.67238	0.613087	0.306543	−	0.951857i	$-0.400828\pi$
$19$	2.67238	0.613087	0.306543	+	0.951857i	$0.400828\pi$
$20$	8.23305	1.84097
$21$	0	0
$22$	−2.89270	−0.616725
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	2.33804	0.467607
$26$	−10.1876	−1.99796
$27$	0	0
$28$	−5.51756	−1.04272
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−5.79860	−1.04146	−0.520730	−	0.853722i	$-0.674340\pi$
$31$	−5.79860	−1.04146	−0.520730	+	0.853722i	$0.674340\pi$
$32$	6.55469	1.15872
$33$	0	0
$34$	10.0702	1.72703
$35$	−4.91775	−0.831251
$36$	0	0
$37$	−5.91943	−0.973147	−0.486574	−	0.873640i	$-0.661754\pi$
$37$	−5.91943	−0.973147	−0.486574	+	0.873640i	$0.661754\pi$
$38$	−5.99906	−0.973175
$39$	0	0
$40$	−6.31985	−0.999255
$41$	−7.82902	−1.22269	−0.611344	−	0.791365i	$-0.709371\pi$
$41$	−7.82902	−1.22269	−0.611344	+	0.791365i	$0.709371\pi$
$42$	0	0
$43$	−5.28362	−0.805744	−0.402872	−	0.915256i	$-0.631988\pi$
$43$	−5.28362	−0.805744	−0.402872	+	0.915256i	$0.631988\pi$
$44$	3.91642	0.590423
$45$	0	0
$46$	−2.24483	−0.330983
$47$	2.52885	0.368871	0.184436	−	0.982845i	$-0.440954\pi$
$47$	2.52885	0.368871	0.184436	+	0.982845i	$0.440954\pi$
$48$	0	0
$49$	−3.70427	−0.529181
$50$	−5.24851	−0.742251
$51$	0	0
$52$	13.7930	1.91275
$53$	7.14230	0.981070	0.490535	−	0.871421i	$-0.336801\pi$
$53$	7.14230	0.981070	0.490535	+	0.871421i	$0.336801\pi$
$54$	0	0
$55$	3.49067	0.470682
$56$	4.23538	0.565977
$57$	0	0
$58$	−2.24483	−0.294761
$59$	−7.87478	−1.02521	−0.512605	−	0.858625i	$-0.671319\pi$
$59$	−7.87478	−1.02521	−0.512605	+	0.858625i	$0.671319\pi$
$60$	0	0
$61$	0.0192849	0.00246918	0.00123459	−	0.999999i	$-0.499607\pi$
$61$	0.0192849	0.00246918	0.00123459	+	0.999999i	$0.499607\pi$
$62$	13.0169	1.65315
$63$	0	0
$64$	−13.0315	−1.62894
$65$	12.2936	1.52483
$66$	0	0
$67$	−15.5222	−1.89634	−0.948170	−	0.317764i	$-0.897068\pi$
$67$	−15.5222	−1.89634	−0.948170	+	0.317764i	$0.897068\pi$
$68$	−13.6340	−1.65337
$69$	0	0
$70$	11.0395	1.31948
$71$	8.18804	0.971742	0.485871	−	0.874031i	$-0.338502\pi$
$71$	8.18804	0.971742	0.485871	+	0.874031i	$0.338502\pi$
$72$	0	0
$73$	−16.8537	−1.97258	−0.986288	−	0.165035i	$-0.947226\pi$
$73$	−16.8537	−1.97258	−0.986288	+	0.165035i	$0.947226\pi$
$74$	13.2881	1.54471
$75$	0	0
$76$	8.12212	0.931671
$77$	−2.33935	−0.266593
$78$	0	0
$79$	2.32730	0.261842	0.130921	−	0.991393i	$-0.458207\pi$
$79$	2.32730	0.261842	0.130921	+	0.991393i	$0.458207\pi$
$80$	−2.27909	−0.254810
$81$	0	0
$82$	17.5749	1.94082
$83$	8.65460	0.949966	0.474983	−	0.879995i	$-0.342454\pi$
$83$	8.65460	0.949966	0.474983	+	0.879995i	$0.342454\pi$
$84$	0	0
$85$	−12.1519	−1.31806
$86$	11.8608	1.27899
$87$	0	0
$88$	−3.00632	−0.320475
$89$	17.6714	1.87316	0.936581	−	0.350451i	$-0.113972\pi$
$89$	17.6714	1.87316	0.936581	+	0.350451i	$0.113972\pi$
$90$	0	0
$91$	−8.23883	−0.863664
$92$	3.03928	0.316867
$93$	0	0
$94$	−5.67686	−0.585523
$95$	7.23917	0.742723
$96$	0	0
$97$	−7.50809	−0.762331	−0.381166	−	0.924507i	$-0.624477\pi$
$97$	−7.50809	−0.762331	−0.381166	+	0.924507i	$0.624477\pi$
$98$	8.31546	0.839989
$99$	0	0
$100$	7.10595	0.710595
$101$	−12.9202	−1.28561	−0.642804	−	0.766031i	$-0.722229\pi$
$101$	−12.9202	−1.28561	−0.642804	+	0.766031i	$0.722229\pi$
$102$	0	0
$103$	0.730126	0.0719415	0.0359707	−	0.999353i	$-0.488548\pi$
$103$	0.730126	0.0719415	0.0359707	+	0.999353i	$0.488548\pi$
$104$	−10.5878	−1.03822
$105$	0	0
$106$	−16.0333	−1.55729
$107$	−2.51770	−0.243395	−0.121697	−	0.992567i	$-0.538834\pi$
$107$	−2.51770	−0.243395	−0.121697	+	0.992567i	$0.538834\pi$
$108$	0	0
$109$	0.164640	0.0157696	0.00788481	−	0.999969i	$-0.497490\pi$
$109$	0.164640	0.0157696	0.00788481	+	0.999969i	$0.497490\pi$
$110$	−7.83597	−0.747131
$111$	0	0
$112$	1.52738	0.144324
$113$	−12.0351	−1.13216	−0.566082	−	0.824349i	$-0.691541\pi$
$113$	−12.0351	−1.13216	−0.566082	+	0.824349i	$0.691541\pi$
$114$	0	0
$115$	2.70888	0.252604
$116$	3.03928	0.282190
$117$	0	0
$118$	17.6776	1.62735
$119$	8.14386	0.746546
$120$	0	0
$121$	−9.33951	−0.849046
$122$	−0.0432915	−0.00391942
$123$	0	0
$124$	−17.6236	−1.58264
$125$	−7.21094	−0.644966
$126$	0	0
$127$	−12.8939	−1.14415	−0.572076	−	0.820201i	$-0.693862\pi$
$127$	−12.8939	−1.14415	−0.572076	+	0.820201i	$0.693862\pi$
$128$	16.1442	1.42696
$129$	0	0
$130$	−27.5971	−2.42043
$131$	−15.3953	−1.34510	−0.672549	−	0.740053i	$-0.734800\pi$
$131$	−15.3953	−1.34510	−0.672549	+	0.740053i	$0.734800\pi$
$132$	0	0
$133$	−4.85149	−0.420677
$134$	34.8448	3.01013
$135$	0	0
$136$	10.4658	0.897431
$137$	−4.85694	−0.414956	−0.207478	−	0.978240i	$-0.566526\pi$
$137$	−4.85694	−0.414956	−0.207478	+	0.978240i	$0.566526\pi$
$138$	0	0
$139$	8.76830	0.743718	0.371859	−	0.928289i	$-0.378721\pi$
$139$	8.76830	0.743718	0.371859	+	0.928289i	$0.378721\pi$
$140$	−14.9464	−1.26320
$141$	0	0
$142$	−18.3808	−1.54248
$143$	5.84801	0.489035
$144$	0	0
$145$	2.70888	0.224960
$146$	37.8337	3.13114
$147$	0	0
$148$	−17.9908	−1.47883
$149$	−1.66660	−0.136534	−0.0682668	−	0.997667i	$-0.521747\pi$
$149$	−1.66660	−0.136534	−0.0682668	+	0.997667i	$0.521747\pi$
$150$	0	0
$151$	15.6639	1.27471	0.637357	−	0.770569i	$-0.280028\pi$
$151$	15.6639	1.27471	0.637357	+	0.770569i	$0.280028\pi$
$152$	−6.23469	−0.505701
$153$	0	0
$154$	5.25145	0.423174
$155$	−15.7077	−1.26167
$156$	0	0
$157$	−17.5561	−1.40113	−0.700563	−	0.713591i	$-0.747068\pi$
$157$	−17.5561	−1.40113	−0.700563	+	0.713591i	$0.747068\pi$
$158$	−5.22441	−0.415632
$159$	0	0
$160$	17.7559	1.40372
$161$	−1.81542	−0.143075
$162$	0	0
$163$	22.2450	1.74236	0.871180	−	0.490964i	$-0.163355\pi$
$163$	22.2450	1.74236	0.871180	+	0.490964i	$0.163355\pi$
$164$	−23.7946	−1.85805
$165$	0	0
$166$	−19.4281	−1.50792
$167$	−20.9725	−1.62290	−0.811449	−	0.584423i	$-0.801321\pi$
$167$	−20.9725	−1.62290	−0.811449	+	0.584423i	$0.801321\pi$
$168$	0	0
$169$	7.59580	0.584292
$170$	27.2790	2.09220
$171$	0	0
$172$	−16.0584	−1.22444
$173$	19.8975	1.51278	0.756389	−	0.654122i	$-0.226962\pi$
$173$	19.8975	1.51278	0.756389	+	0.654122i	$0.226962\pi$
$174$	0	0
$175$	−4.24451	−0.320855
$176$	−1.08415	−0.0817210
$177$	0	0
$178$	−39.6693	−2.97334
$179$	0.697223	0.0521129	0.0260564	−	0.999660i	$-0.491705\pi$
$179$	0.697223	0.0521129	0.0260564	+	0.999660i	$0.491705\pi$
$180$	0	0
$181$	−21.7053	−1.61334	−0.806670	−	0.591001i	$-0.798733\pi$
$181$	−21.7053	−1.61334	−0.806670	+	0.591001i	$0.798733\pi$
$182$	18.4948	1.37093
$183$	0	0
$184$	−2.33301	−0.171992
$185$	−16.0350	−1.17892
$186$	0	0
$187$	−5.78060	−0.422719
$188$	7.68589	0.560551
$189$	0	0
$190$	−16.2507	−1.17895
$191$	14.0930	1.01974	0.509869	−	0.860252i	$-0.329694\pi$
$191$	14.0930	1.01974	0.509869	+	0.860252i	$0.329694\pi$
$192$	0	0
$193$	11.1691	0.803973	0.401986	−	0.915646i	$-0.368320\pi$
$193$	11.1691	0.803973	0.401986	+	0.915646i	$0.368320\pi$
$194$	16.8544	1.21008
$195$	0	0
$196$	−11.2583	−0.804164
$197$	11.8308	0.842910	0.421455	−	0.906849i	$-0.361520\pi$
$197$	11.8308	0.842910	0.421455	+	0.906849i	$0.361520\pi$
$198$	0	0
$199$	7.23419	0.512818	0.256409	−	0.966568i	$-0.417461\pi$
$199$	7.23419	0.512818	0.256409	+	0.966568i	$0.417461\pi$
$200$	−5.45466	−0.385703
$201$	0	0
$202$	29.0037	2.04069
$203$	−1.81542	−0.127417
$204$	0	0
$205$	−21.2079	−1.48122
$206$	−1.63901	−0.114195
$207$	0	0
$208$	−3.81822	−0.264746
$209$	3.44364	0.238201
$210$	0	0
$211$	−4.94613	−0.340506	−0.170253	−	0.985400i	$-0.554458\pi$
$211$	−4.94613	−0.340506	−0.170253	+	0.985400i	$0.554458\pi$
$212$	21.7074	1.49087
$213$	0	0
$214$	5.65181	0.386350
$215$	−14.3127	−0.976117
$216$	0	0
$217$	10.5269	0.714610
$218$	−0.369589	−0.0250317
$219$	0	0
$220$	10.6091	0.715267
$221$	−20.3584	−1.36945
$222$	0	0
$223$	−12.4975	−0.836891	−0.418446	−	0.908242i	$-0.637425\pi$
$223$	−12.4975	−0.836891	−0.418446	+	0.908242i	$0.637425\pi$
$224$	−11.8995	−0.795068
$225$	0	0
$226$	27.0167	1.79712
$227$	−14.7488	−0.978911	−0.489456	−	0.872028i	$-0.662804\pi$
$227$	−14.7488	−0.978911	−0.489456	+	0.872028i	$0.662804\pi$
$228$	0	0
$229$	−11.4103	−0.754016	−0.377008	−	0.926210i	$-0.623047\pi$
$229$	−11.4103	−0.754016	−0.377008	+	0.926210i	$0.623047\pi$
$230$	−6.08099	−0.400969
$231$	0	0
$232$	−2.33301	−0.153170
$233$	−10.5481	−0.691029	−0.345514	−	0.938413i	$-0.612296\pi$
$233$	−10.5481	−0.691029	−0.345514	+	0.938413i	$0.612296\pi$
$234$	0	0
$235$	6.85036	0.446868
$236$	−23.9337	−1.55795
$237$	0	0
$238$	−18.2816	−1.18502
$239$	−3.76141	−0.243306	−0.121653	−	0.992573i	$-0.538819\pi$
$239$	−3.76141	−0.243306	−0.121653	+	0.992573i	$0.538819\pi$
$240$	0	0
$241$	−11.4984	−0.740677	−0.370339	−	0.928897i	$-0.620758\pi$
$241$	−11.4984	−0.740677	−0.370339	+	0.928897i	$0.620758\pi$
$242$	20.9656	1.34772
$243$	0	0
$244$	0.0586123	0.00375227
$245$	−10.0344	−0.641075
$246$	0	0
$247$	12.1280	0.771684
$248$	13.5282	0.859041
$249$	0	0
$250$	16.1874	1.02378
$251$	1.24214	0.0784034	0.0392017	−	0.999231i	$-0.487519\pi$
$251$	1.24214	0.0784034	0.0392017	+	0.999231i	$0.487519\pi$
$252$	0	0
$253$	1.28860	0.0810137
$254$	28.9447	1.81615
$255$	0	0
$256$	−10.1780	−0.636126
$257$	11.4715	0.715575	0.357788	−	0.933803i	$-0.383531\pi$
$257$	11.4715	0.715575	0.357788	+	0.933803i	$0.383531\pi$
$258$	0	0
$259$	10.7462	0.667737
$260$	37.3637	2.31720
$261$	0	0
$262$	34.5600	2.13512
$263$	31.2502	1.92697	0.963486	−	0.267760i	$-0.0862836\pi$
$263$	31.2502	1.92697	0.963486	+	0.267760i	$0.0862836\pi$
$264$	0	0
$265$	19.3476	1.18852
$266$	10.8908	0.667757
$267$	0	0
$268$	−47.1763	−2.88175
$269$	24.6573	1.50338	0.751691	−	0.659516i	$-0.229239\pi$
$269$	24.6573	1.50338	0.751691	+	0.659516i	$0.229239\pi$
$270$	0	0
$271$	26.4463	1.60650	0.803248	−	0.595644i	$-0.203103\pi$
$271$	26.4463	1.60650	0.803248	+	0.595644i	$0.203103\pi$
$272$	3.77421	0.228845
$273$	0	0
$274$	10.9030	0.658675
$275$	3.01280	0.181679
$276$	0	0
$277$	17.9981	1.08140	0.540702	−	0.841214i	$-0.318159\pi$
$277$	17.9981	1.08140	0.540702	+	0.841214i	$0.318159\pi$
$278$	−19.6834	−1.18053
$279$	0	0
$280$	11.4731	0.685652
$281$	−19.6129	−1.17001	−0.585004	−	0.811030i	$-0.698907\pi$
$281$	−19.6129	−1.17001	−0.585004	+	0.811030i	$0.698907\pi$
$282$	0	0
$283$	4.74374	0.281986	0.140993	−	0.990011i	$-0.454971\pi$
$283$	4.74374	0.281986	0.140993	+	0.990011i	$0.454971\pi$
$284$	24.8858	1.47670
$285$	0	0
$286$	−13.1278	−0.776264
$287$	14.2129	0.838963
$288$	0	0
$289$	3.12372	0.183749
$290$	−6.08099	−0.357088
$291$	0	0
$292$	−51.2231	−2.99760
$293$	5.47082	0.319609	0.159804	−	0.987149i	$-0.448914\pi$
$293$	5.47082	0.319609	0.159804	+	0.987149i	$0.448914\pi$
$294$	0	0
$295$	−21.3319	−1.24199
$296$	13.8101	0.802694
$297$	0	0
$298$	3.74125	0.216725
$299$	4.53826	0.262454
$300$	0	0
$301$	9.59196	0.552871
$302$	−35.1629	−2.02340
$303$	0	0
$304$	−2.24838	−0.128954
$305$	0.0522406	0.00299129
$306$	0	0
$307$	−29.4112	−1.67858	−0.839292	−	0.543681i	$-0.817030\pi$
$307$	−29.4112	−1.67858	−0.839292	+	0.543681i	$0.817030\pi$
$308$	−7.10993	−0.405126
$309$	0	0
$310$	35.2612	2.00270
$311$	0.290134	0.0164520	0.00822598	−	0.999966i	$-0.497382\pi$
$311$	0.290134	0.0164520	0.00822598	+	0.999966i	$0.497382\pi$
$312$	0	0
$313$	2.75393	0.155661	0.0778306	−	0.996967i	$-0.475201\pi$
$313$	2.75393	0.155661	0.0778306	+	0.996967i	$0.475201\pi$
$314$	39.4104	2.22406
$315$	0	0
$316$	7.07332	0.397905
$317$	−23.3102	−1.30923	−0.654617	−	0.755961i	$-0.727170\pi$
$317$	−23.3102	−1.30923	−0.654617	+	0.755961i	$0.727170\pi$
$318$	0	0
$319$	1.28860	0.0721478
$320$	−35.3008	−1.97337
$321$	0	0
$322$	4.07531	0.227108
$323$	−11.9882	−0.667040
$324$	0	0
$325$	10.6106	0.588571
$326$	−49.9363	−2.76571
$327$	0	0
$328$	18.2652	1.00853
$329$	−4.59092	−0.253106
$330$	0	0
$331$	28.6297	1.57363	0.786816	−	0.617187i	$-0.211728\pi$
$331$	28.6297	1.57363	0.786816	+	0.617187i	$0.211728\pi$
$332$	26.3038	1.44361
$333$	0	0
$334$	47.0797	2.57609
$335$	−42.0478	−2.29732
$336$	0	0
$337$	−23.8515	−1.29927	−0.649637	−	0.760245i	$-0.725079\pi$
$337$	−23.8515	−1.29927	−0.649637	+	0.760245i	$0.725079\pi$
$338$	−17.0513	−0.927469
$339$	0	0
$340$	−36.9330	−2.00297
$341$	−7.47208	−0.404636
$342$	0	0
$343$	19.4327	1.04927
$344$	12.3267	0.664613
$345$	0	0
$346$	−44.6666	−2.40129
$347$	12.6128	0.677093	0.338546	−	0.940950i	$-0.390065\pi$
$347$	12.6128	0.677093	0.338546	+	0.940950i	$0.390065\pi$
$348$	0	0
$349$	28.6656	1.53443	0.767216	−	0.641389i	$-0.221641\pi$
$349$	28.6656	1.53443	0.767216	+	0.641389i	$0.221641\pi$
$350$	9.52822	0.509305
$351$	0	0
$352$	8.44638	0.450194
$353$	9.85003	0.524264	0.262132	−	0.965032i	$-0.415574\pi$
$353$	9.85003	0.524264	0.262132	+	0.965032i	$0.415574\pi$
$354$	0	0
$355$	22.1804	1.17722
$356$	53.7082	2.84653
$357$	0	0
$358$	−1.56515	−0.0827207
$359$	−6.28877	−0.331909	−0.165954	−	0.986133i	$-0.553070\pi$
$359$	−6.28877	−0.331909	−0.165954	+	0.986133i	$0.553070\pi$
$360$	0	0
$361$	−11.8584	−0.624125
$362$	48.7248	2.56092
$363$	0	0
$364$	−25.0401	−1.31246
$365$	−45.6546	−2.38967
$366$	0	0
$367$	34.2153	1.78602	0.893011	−	0.450034i	$-0.148588\pi$
$367$	34.2153	1.78602	0.893011	+	0.450034i	$0.148588\pi$
$368$	−0.841340	−0.0438579
$369$	0	0
$370$	35.9960	1.87134
$371$	−12.9662	−0.673174
$372$	0	0
$373$	6.32053	0.327264	0.163632	−	0.986521i	$-0.447679\pi$
$373$	6.32053	0.327264	0.163632	+	0.986521i	$0.447679\pi$
$374$	12.9765	0.670998
$375$	0	0
$376$	−5.89984	−0.304261
$377$	4.53826	0.233732
$378$	0	0
$379$	24.3916	1.25291	0.626456	−	0.779457i	$-0.284505\pi$
$379$	24.3916	1.25291	0.626456	+	0.779457i	$0.284505\pi$
$380$	22.0019	1.12867
$381$	0	0
$382$	−31.6366	−1.61867
$383$	1.48836	0.0760516	0.0380258	−	0.999277i	$-0.487893\pi$
$383$	1.48836	0.0760516	0.0380258	+	0.999277i	$0.487893\pi$
$384$	0	0
$385$	−6.33701	−0.322964
$386$	−25.0729	−1.27618
$387$	0	0
$388$	−22.8192	−1.15847
$389$	32.8601	1.66607	0.833036	−	0.553219i	$-0.186601\pi$
$389$	32.8601	1.66607	0.833036	+	0.553219i	$0.186601\pi$
$390$	0	0
$391$	−4.48595	−0.226864
$392$	8.64209	0.436491
$393$	0	0
$394$	−26.5582	−1.33798
$395$	6.30439	0.317208
$396$	0	0
$397$	6.95067	0.348844	0.174422	−	0.984671i	$-0.444194\pi$
$397$	6.95067	0.348844	0.174422	+	0.984671i	$0.444194\pi$
$398$	−16.2396	−0.814015
$399$	0	0
$400$	−1.96708	−0.0983542
$401$	−2.64996	−0.132333	−0.0661664	−	0.997809i	$-0.521077\pi$
$401$	−2.64996	−0.132333	−0.0661664	+	0.997809i	$0.521077\pi$
$402$	0	0
$403$	−26.3155	−1.31087
$404$	−39.2681	−1.95366
$405$	0	0
$406$	4.07531	0.202254
$407$	−7.62778	−0.378095
$408$	0	0
$409$	−8.22912	−0.406904	−0.203452	−	0.979085i	$-0.565216\pi$
$409$	−8.22912	−0.406904	−0.203452	+	0.979085i	$0.565216\pi$
$410$	47.6082	2.35120
$411$	0	0
$412$	2.21906	0.109325
$413$	14.2960	0.703460
$414$	0	0
$415$	23.4443	1.15083
$416$	29.7469	1.45846
$417$	0	0
$418$	−7.73039	−0.378106
$419$	−17.7332	−0.866323	−0.433161	−	0.901316i	$-0.642602\pi$
$419$	−17.7332	−0.866323	−0.433161	+	0.901316i	$0.642602\pi$
$420$	0	0
$421$	19.8575	0.967795	0.483897	−	0.875125i	$-0.339221\pi$
$421$	19.8575	0.967795	0.483897	+	0.875125i	$0.339221\pi$
$422$	11.1032	0.540497
$423$	0	0
$424$	−16.6631	−0.809229
$425$	−10.4883	−0.508758
$426$	0	0
$427$	−0.0350102	−0.00169426
$428$	−7.65198	−0.369873
$429$	0	0
$430$	32.1296	1.54943
$431$	16.6485	0.801930	0.400965	−	0.916093i	$-0.368675\pi$
$431$	16.6485	0.801930	0.400965	+	0.916093i	$0.368675\pi$
$432$	0	0
$433$	−33.5474	−1.61219	−0.806093	−	0.591789i	$-0.798422\pi$
$433$	−33.5474	−1.61219	−0.806093	+	0.591789i	$0.798422\pi$
$434$	−23.6311	−1.13433
$435$	0	0
$436$	0.500386	0.0239641
$437$	2.67238	0.127837
$438$	0	0
$439$	18.1685	0.867134	0.433567	−	0.901121i	$-0.357255\pi$
$439$	18.1685	0.867134	0.433567	+	0.901121i	$0.357255\pi$
$440$	−8.14376	−0.388239
$441$	0	0
$442$	45.7012	2.17379
$443$	−26.5362	−1.26077	−0.630386	−	0.776282i	$-0.717103\pi$
$443$	−26.5362	−1.26077	−0.630386	+	0.776282i	$0.717103\pi$
$444$	0	0
$445$	47.8697	2.26924
$446$	28.0547	1.32843
$447$	0	0
$448$	23.6576	1.11772
$449$	−31.8845	−1.50472	−0.752362	−	0.658750i	$-0.771085\pi$
$449$	−31.8845	−1.50472	−0.752362	+	0.658750i	$0.771085\pi$
$450$	0	0
$451$	−10.0885	−0.475049
$452$	−36.5779	−1.72048
$453$	0	0
$454$	33.1086	1.55386
$455$	−22.3180	−1.04628
$456$	0	0
$457$	−3.81457	−0.178438	−0.0892190	−	0.996012i	$-0.528437\pi$
$457$	−3.81457	−0.178438	−0.0892190	+	0.996012i	$0.528437\pi$
$458$	25.6143	1.19688
$459$	0	0
$460$	8.23305	0.383868
$461$	15.5993	0.726532	0.363266	−	0.931685i	$-0.381662\pi$
$461$	15.5993	0.726532	0.363266	+	0.931685i	$0.381662\pi$
$462$	0	0
$463$	−32.0523	−1.48960	−0.744799	−	0.667289i	$-0.767455\pi$
$463$	−32.0523	−1.48960	−0.744799	+	0.667289i	$0.767455\pi$
$464$	−0.841340	−0.0390582
$465$	0	0
$466$	23.6787	1.09690
$467$	−27.7274	−1.28307	−0.641535	−	0.767094i	$-0.721702\pi$
$467$	−27.7274	−1.28307	−0.641535	+	0.767094i	$0.721702\pi$
$468$	0	0
$469$	28.1793	1.30120
$470$	−15.3779	−0.709331
$471$	0	0
$472$	18.3719	0.845637
$473$	−6.80848	−0.313054
$474$	0	0
$475$	6.24813	0.286684
$476$	24.7515	1.13448
$477$	0	0
$478$	8.44375	0.386208
$479$	−28.8506	−1.31822	−0.659109	−	0.752048i	$-0.729066\pi$
$479$	−28.8506	−1.31822	−0.659109	+	0.752048i	$0.729066\pi$
$480$	0	0
$481$	−26.8639	−1.22489
$482$	25.8120	1.17570
$483$	0	0
$484$	−28.3854	−1.29024
$485$	−20.3385	−0.923525
$486$	0	0
$487$	−13.0525	−0.591467	−0.295734	−	0.955270i	$-0.595564\pi$
$487$	−13.0525	−0.591467	−0.295734	+	0.955270i	$0.595564\pi$
$488$	−0.0449919	−0.00203669
$489$	0	0
$490$	22.5256	1.01760
$491$	16.7180	0.754472	0.377236	−	0.926117i	$-0.376875\pi$
$491$	16.7180	0.754472	0.377236	+	0.926117i	$0.376875\pi$
$492$	0	0
$493$	−4.48595	−0.202037
$494$	−27.2253	−1.22492
$495$	0	0
$496$	4.87859	0.219055
$497$	−14.8647	−0.666773
$498$	0	0
$499$	5.64497	0.252704	0.126352	−	0.991985i	$-0.459673\pi$
$499$	5.64497	0.252704	0.126352	+	0.991985i	$0.459673\pi$
$500$	−21.9161	−0.980116
$501$	0	0
$502$	−2.78841	−0.124453
$503$	−18.7912	−0.837858	−0.418929	−	0.908019i	$-0.637594\pi$
$503$	−18.7912	−0.837858	−0.418929	+	0.908019i	$0.637594\pi$
$504$	0	0
$505$	−34.9993	−1.55745
$506$	−2.89270	−0.128596
$507$	0	0
$508$	−39.1883	−1.73870
$509$	38.1542	1.69116	0.845579	−	0.533851i	$-0.179256\pi$
$509$	38.1542	1.69116	0.845579	+	0.533851i	$0.179256\pi$
$510$	0	0
$511$	30.5965	1.35351
$512$	−9.44042	−0.417212
$513$	0	0
$514$	−25.7517	−1.13586
$515$	1.97783	0.0871534
$516$	0	0
$517$	3.25868	0.143317
$518$	−24.1235	−1.05992
$519$	0	0
$520$	−28.6811	−1.25775
$521$	20.7383	0.908564	0.454282	−	0.890858i	$-0.349896\pi$
$521$	20.7383	0.908564	0.454282	+	0.890858i	$0.349896\pi$
$522$	0	0
$523$	−35.6601	−1.55931	−0.779654	−	0.626211i	$-0.784605\pi$
$523$	−35.6601	−1.55931	−0.779654	+	0.626211i	$0.784605\pi$
$524$	−46.7907	−2.04406
$525$	0	0
$526$	−70.1516	−3.05875
$527$	26.0122	1.13311
$528$	0	0
$529$	1.00000	0.0434783
$530$	−43.4322	−1.88658
$531$	0	0
$532$	−14.7450	−0.639278
$533$	−35.5301	−1.53898
$534$	0	0
$535$	−6.82014	−0.294860
$536$	36.2135	1.56418
$537$	0	0
$538$	−55.3515	−2.38637
$539$	−4.77332	−0.205602
$540$	0	0
$541$	−12.9936	−0.558640	−0.279320	−	0.960198i	$-0.590109\pi$
$541$	−12.9936	−0.558640	−0.279320	+	0.960198i	$0.590109\pi$
$542$	−59.3675	−2.55005
$543$	0	0
$544$	−29.4040	−1.26069
$545$	0.445989	0.0191041
$546$	0	0
$547$	35.0835	1.50006	0.750031	−	0.661403i	$-0.230039\pi$
$547$	35.0835	1.50006	0.750031	+	0.661403i	$0.230039\pi$
$548$	−14.7616	−0.630584
$549$	0	0
$550$	−6.76323	−0.288385
$551$	2.67238	0.113847
$552$	0	0
$553$	−4.22502	−0.179666
$554$	−40.4028	−1.71655
$555$	0	0
$556$	26.6493	1.13018
$557$	38.1951	1.61838	0.809190	−	0.587547i	$-0.199906\pi$
$557$	38.1951	1.61838	0.809190	+	0.587547i	$0.199906\pi$
$558$	0	0
$559$	−23.9784	−1.01418
$560$	4.13749	0.174841
$561$	0	0
$562$	44.0277	1.85720
$563$	11.2679	0.474886	0.237443	−	0.971401i	$-0.423691\pi$
$563$	11.2679	0.474886	0.237443	+	0.971401i	$0.423691\pi$
$564$	0	0
$565$	−32.6015	−1.37156
$566$	−10.6489	−0.447607
$567$	0	0
$568$	−19.1028	−0.801535
$569$	−21.5961	−0.905356	−0.452678	−	0.891674i	$-0.649531\pi$
$569$	−21.5961	−0.905356	−0.452678	+	0.891674i	$0.649531\pi$
$570$	0	0
$571$	8.89694	0.372325	0.186163	−	0.982519i	$-0.440395\pi$
$571$	8.89694	0.372325	0.186163	+	0.982519i	$0.440395\pi$
$572$	17.7737	0.743157
$573$	0	0
$574$	−31.9057	−1.33172
$575$	2.33804	0.0975029
$576$	0	0
$577$	−29.9448	−1.24662	−0.623308	−	0.781976i	$-0.714212\pi$
$577$	−29.9448	−1.24662	−0.623308	+	0.781976i	$0.714212\pi$
$578$	−7.01224	−0.291671
$579$	0	0
$580$	8.23305	0.341859
$581$	−15.7117	−0.651831
$582$	0	0
$583$	9.20358	0.381173
$584$	39.3198	1.62707
$585$	0	0
$586$	−12.2811	−0.507327
$587$	−23.3343	−0.963109	−0.481555	−	0.876416i	$-0.659928\pi$
$587$	−23.3343	−0.963109	−0.481555	+	0.876416i	$0.659928\pi$
$588$	0	0
$589$	−15.4961	−0.638505
$590$	47.8865	1.97145
$591$	0	0
$592$	4.98025	0.204687
$593$	−34.8623	−1.43162	−0.715811	−	0.698295i	$-0.753943\pi$
$593$	−34.8623	−1.43162	−0.715811	+	0.698295i	$0.753943\pi$
$594$	0	0
$595$	22.0607	0.904402
$596$	−5.06528	−0.207482
$597$	0	0
$598$	−10.1876	−0.416604
$599$	−13.7736	−0.562775	−0.281387	−	0.959594i	$-0.590795\pi$
$599$	−13.7736	−0.562775	−0.281387	+	0.959594i	$0.590795\pi$
$600$	0	0
$601$	−11.6882	−0.476772	−0.238386	−	0.971170i	$-0.576618\pi$
$601$	−11.6882	−0.476772	−0.238386	+	0.971170i	$0.576618\pi$
$602$	−21.5324	−0.877593
$603$	0	0
$604$	47.6071	1.93710
$605$	−25.2996	−1.02858
$606$	0	0
$607$	−22.5491	−0.915238	−0.457619	−	0.889148i	$-0.651298\pi$
$607$	−22.5491	−0.915238	−0.457619	+	0.889148i	$0.651298\pi$
$608$	17.5166	0.710393
$609$	0	0
$610$	−0.117271	−0.00474818
$611$	11.4766	0.464293
$612$	0	0
$613$	−30.0844	−1.21510	−0.607549	−	0.794282i	$-0.707847\pi$
$613$	−30.0844	−1.21510	−0.607549	+	0.794282i	$0.707847\pi$
$614$	66.0232	2.66448
$615$	0	0
$616$	5.45772	0.219898
$617$	−3.34573	−0.134694	−0.0673471	−	0.997730i	$-0.521453\pi$
$617$	−3.34573	−0.134694	−0.0673471	+	0.997730i	$0.521453\pi$
$618$	0	0
$619$	−27.0042	−1.08539	−0.542694	−	0.839930i	$-0.682596\pi$
$619$	−27.0042	−1.08539	−0.542694	+	0.839930i	$0.682596\pi$
$620$	−47.7401	−1.91729
$621$	0	0
$622$	−0.651302	−0.0261148
$623$	−32.0809	−1.28529
$624$	0	0
$625$	−31.2238	−1.24895
$626$	−6.18211	−0.247087
$627$	0	0
$628$	−53.3578	−2.12921
$629$	26.5542	1.05879
$630$	0	0
$631$	17.3154	0.689316	0.344658	−	0.938728i	$-0.387995\pi$
$631$	17.3154	0.689316	0.344658	+	0.938728i	$0.387995\pi$
$632$	−5.42962	−0.215979
$633$	0	0
$634$	52.3276	2.07819
$635$	−34.9281	−1.38608
$636$	0	0
$637$	−16.8109	−0.666073
$638$	−2.89270	−0.114523
$639$	0	0
$640$	43.7327	1.72869
$641$	−48.1364	−1.90128	−0.950638	−	0.310304i	$-0.899569\pi$
$641$	−48.1364	−1.90128	−0.950638	+	0.310304i	$0.899569\pi$
$642$	0	0
$643$	−33.4951	−1.32092	−0.660460	−	0.750862i	$-0.729639\pi$
$643$	−33.4951	−1.32092	−0.660460	+	0.750862i	$0.729639\pi$
$644$	−5.51756	−0.217422
$645$	0	0
$646$	26.9114	1.05882
$647$	−15.8951	−0.624900	−0.312450	−	0.949934i	$-0.601150\pi$
$647$	−15.8951	−0.624900	−0.312450	+	0.949934i	$0.601150\pi$
$648$	0	0
$649$	−10.1475	−0.398323
$650$	−23.8191	−0.934261
$651$	0	0
$652$	67.6087	2.64776
$653$	−28.6078	−1.11951	−0.559754	−	0.828659i	$-0.689104\pi$
$653$	−28.6078	−1.11951	−0.559754	+	0.828659i	$0.689104\pi$
$654$	0	0
$655$	−41.7041	−1.62952
$656$	6.58687	0.257174
$657$	0	0
$658$	10.3059	0.401764
$659$	32.2194	1.25509	0.627545	−	0.778580i	$-0.284060\pi$
$659$	32.2194	1.25509	0.627545	+	0.778580i	$0.284060\pi$
$660$	0	0
$661$	−7.50116	−0.291761	−0.145881	−	0.989302i	$-0.546602\pi$
$661$	−7.50116	−0.291761	−0.145881	+	0.989302i	$0.546602\pi$
$662$	−64.2690	−2.49789
$663$	0	0
$664$	−20.1913	−0.783573
$665$	−13.1421	−0.509629
$666$	0	0
$667$	1.00000	0.0387202
$668$	−63.7412	−2.46622
$669$	0	0
$670$	94.3904	3.64662
$671$	0.0248506	0.000959346 0
$672$	0	0
$673$	13.3328	0.513941	0.256970	−	0.966419i	$-0.417276\pi$
$673$	13.3328	0.513941	0.256970	+	0.966419i	$0.417276\pi$
$674$	53.5426	2.06239
$675$	0	0
$676$	23.0858	0.887914
$677$	25.8323	0.992817	0.496408	−	0.868089i	$-0.334652\pi$
$677$	25.8323	0.992817	0.496408	+	0.868089i	$0.334652\pi$
$678$	0	0
$679$	13.6303	0.523083
$680$	28.3505	1.08719
$681$	0	0
$682$	16.7736	0.642294
$683$	2.67992	0.102544	0.0512722	−	0.998685i	$-0.483672\pi$
$683$	2.67992	0.102544	0.0512722	+	0.998685i	$0.483672\pi$
$684$	0	0
$685$	−13.1569	−0.502698
$686$	−43.6232	−1.66554
$687$	0	0
$688$	4.44532	0.169476
$689$	32.4136	1.23486
$690$	0	0
$691$	−14.5615	−0.553945	−0.276972	−	0.960878i	$-0.589331\pi$
$691$	−14.5615	−0.553945	−0.276972	+	0.960878i	$0.589331\pi$
$692$	60.4740	2.29888
$693$	0	0
$694$	−28.3137	−1.07477
$695$	23.7523	0.900976
$696$	0	0
$697$	35.1206	1.33029
$698$	−64.3494	−2.43566
$699$	0	0
$700$	−12.9003	−0.487584
$701$	−12.8173	−0.484103	−0.242051	−	0.970263i	$-0.577820\pi$
$701$	−12.8173	−0.484103	−0.242051	+	0.970263i	$0.577820\pi$
$702$	0	0
$703$	−15.8190	−0.596624
$704$	−16.7924	−0.632888
$705$	0	0
$706$	−22.1117	−0.832185
$707$	23.4555	0.882136
$708$	0	0
$709$	−10.1202	−0.380073	−0.190036	−	0.981777i	$-0.560861\pi$
$709$	−10.1202	−0.380073	−0.190036	+	0.981777i	$0.560861\pi$
$710$	−49.7914	−1.86864
$711$	0	0
$712$	−41.2275	−1.54507
$713$	−5.79860	−0.217159
$714$	0	0
$715$	15.8416	0.592441
$716$	2.11905	0.0791928
$717$	0	0
$718$	14.1172	0.526851
$719$	50.2292	1.87323	0.936616	−	0.350357i	$-0.113940\pi$
$719$	50.2292	1.87323	0.936616	+	0.350357i	$0.113940\pi$
$720$	0	0
$721$	−1.32548	−0.0493636
$722$	26.6201	0.990697
$723$	0	0
$724$	−65.9684	−2.45170
$725$	2.33804	0.0868325
$726$	0	0
$727$	−30.3374	−1.12515	−0.562575	−	0.826746i	$-0.690189\pi$
$727$	−30.3374	−1.12515	−0.562575	+	0.826746i	$0.690189\pi$
$728$	19.2213	0.712387
$729$	0	0
$730$	102.487	3.79322
$731$	23.7020	0.876651
$732$	0	0
$733$	−29.1824	−1.07788	−0.538939	−	0.842345i	$-0.681175\pi$
$733$	−29.1824	−1.07788	−0.538939	+	0.842345i	$0.681175\pi$
$734$	−76.8076	−2.83502
$735$	0	0
$736$	6.55469	0.241609
$737$	−20.0019	−0.736781
$738$	0	0
$739$	−17.6927	−0.650837	−0.325419	−	0.945570i	$-0.605505\pi$
$739$	−17.6927	−0.650837	−0.325419	+	0.945570i	$0.605505\pi$
$740$	−48.7349	−1.79153
$741$	0	0
$742$	29.1071	1.06855
$743$	−25.7647	−0.945216	−0.472608	−	0.881273i	$-0.656687\pi$
$743$	−25.7647	−0.945216	−0.472608	+	0.881273i	$0.656687\pi$
$744$	0	0
$745$	−4.51463	−0.165403
$746$	−14.1885	−0.519479
$747$	0	0
$748$	−17.5689	−0.642381
$749$	4.57067	0.167008
$750$	0	0
$751$	0.702382	0.0256303	0.0128151	−	0.999918i	$-0.495921\pi$
$751$	0.702382	0.0256303	0.0128151	+	0.999918i	$0.495921\pi$
$752$	−2.12762	−0.0775865
$753$	0	0
$754$	−10.1876	−0.371012
$755$	42.4318	1.54425
$756$	0	0
$757$	−33.7740	−1.22754	−0.613768	−	0.789486i	$-0.710347\pi$
$757$	−33.7740	−1.22754	−0.613768	+	0.789486i	$0.710347\pi$
$758$	−54.7551	−1.98879
$759$	0	0
$760$	−16.8890	−0.612630
$761$	−33.2497	−1.20530	−0.602651	−	0.798005i	$-0.705889\pi$
$761$	−33.2497	−1.20530	−0.602651	+	0.798005i	$0.705889\pi$
$762$	0	0
$763$	−0.298889	−0.0108205
$764$	42.8327	1.54963
$765$	0	0
$766$	−3.34112	−0.120720
$767$	−35.7378	−1.29042
$768$	0	0
$769$	27.8729	1.00512	0.502561	−	0.864542i	$-0.332391\pi$
$769$	27.8729	1.00512	0.502561	+	0.864542i	$0.332391\pi$
$770$	14.2255	0.512653
$771$	0	0
$772$	33.9462	1.22175
$773$	−39.7315	−1.42904	−0.714522	−	0.699613i	$-0.753356\pi$
$773$	−39.7315	−1.42904	−0.714522	+	0.699613i	$0.753356\pi$
$774$	0	0
$775$	−13.5573	−0.486994
$776$	17.5165	0.628804
$777$	0	0
$778$	−73.7654	−2.64462
$779$	−20.9222	−0.749614
$780$	0	0
$781$	10.5511	0.377549
$782$	10.0702	0.360110
$783$	0	0
$784$	3.11655	0.111305
$785$	−47.5573	−1.69739
$786$	0	0
$787$	6.77874	0.241636	0.120818	−	0.992675i	$-0.461448\pi$
$787$	6.77874	0.241636	0.120818	+	0.992675i	$0.461448\pi$
$788$	35.9571	1.28092
$789$	0	0
$790$	−14.1523	−0.503516
$791$	21.8486	0.776848
$792$	0	0
$793$	0.0875200	0.00310793
$794$	−15.6031	−0.553734
$795$	0	0
$796$	21.9867	0.779299
$797$	−45.6601	−1.61737	−0.808683	−	0.588245i	$-0.799819\pi$
$797$	−45.6601	−1.61737	−0.808683	+	0.588245i	$0.799819\pi$
$798$	0	0
$799$	−11.3443	−0.401333
$800$	15.3251	0.541824
$801$	0	0
$802$	5.94872	0.210057
$803$	−21.7177	−0.766401
$804$	0	0
$805$	−4.91775	−0.173328
$806$	59.0740	2.08079
$807$	0	0
$808$	30.1429	1.06042
$809$	30.4752	1.07145	0.535725	−	0.844392i	$-0.320038\pi$
$809$	30.4752	1.07145	0.535725	+	0.844392i	$0.320038\pi$
$810$	0	0
$811$	49.2770	1.73035	0.865174	−	0.501472i	$-0.167208\pi$
$811$	49.2770	1.73035	0.865174	+	0.501472i	$0.167208\pi$
$812$	−5.51756	−0.193628
$813$	0	0
$814$	17.1231	0.600164
$815$	60.2590	2.11078
$816$	0	0
$817$	−14.1198	−0.493991
$818$	18.4730	0.645893
$819$	0	0
$820$	−64.4567	−2.25093
$821$	11.2716	0.393382	0.196691	−	0.980466i	$-0.436980\pi$
$821$	11.2716	0.393382	0.196691	+	0.980466i	$0.436980\pi$
$822$	0	0
$823$	−28.9336	−1.00856	−0.504282	−	0.863539i	$-0.668243\pi$
$823$	−28.9336	−1.00856	−0.504282	+	0.863539i	$0.668243\pi$
$824$	−1.70339	−0.0593405
$825$	0	0
$826$	−32.0922	−1.11663
$827$	28.1166	0.977711	0.488855	−	0.872365i	$-0.337415\pi$
$827$	28.1166	0.977711	0.488855	+	0.872365i	$0.337415\pi$
$828$	0	0
$829$	−24.8841	−0.864262	−0.432131	−	0.901811i	$-0.642238\pi$
$829$	−24.8841	−0.864262	−0.432131	+	0.901811i	$0.642238\pi$
$830$	−52.6285	−1.82676
$831$	0	0
$832$	−59.1404	−2.05032
$833$	16.6171	0.575750
$834$	0	0
$835$	−56.8119	−1.96606
$836$	10.4662	0.361980
$837$	0	0
$838$	39.8080	1.37515
$839$	1.16783	0.0403180	0.0201590	−	0.999797i	$-0.493583\pi$
$839$	1.16783	0.0403180	0.0201590	+	0.999797i	$0.493583\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−44.5768	−1.53622
$843$	0	0
$844$	−15.0327	−0.517446
$845$	20.5761	0.707840
$846$	0	0
$847$	16.9551	0.582584
$848$	−6.00910	−0.206353
$849$	0	0
$850$	23.5445	0.807570
$851$	−5.91943	−0.202915
$852$	0	0
$853$	7.52282	0.257577	0.128788	−	0.991672i	$-0.458891\pi$
$853$	7.52282	0.257577	0.128788	+	0.991672i	$0.458891\pi$
$854$	0.0785920	0.00268936
$855$	0	0
$856$	5.87381	0.200763
$857$	−37.9847	−1.29753	−0.648766	−	0.760988i	$-0.724715\pi$
$857$	−37.9847	−1.29753	−0.648766	+	0.760988i	$0.724715\pi$
$858$	0	0
$859$	−20.5333	−0.700588	−0.350294	−	0.936640i	$-0.613918\pi$
$859$	−20.5333	−0.700588	−0.350294	+	0.936640i	$0.613918\pi$
$860$	−43.5003	−1.48335
$861$	0	0
$862$	−37.3731	−1.27293
$863$	22.2904	0.758773	0.379386	−	0.925238i	$-0.376135\pi$
$863$	22.2904	0.758773	0.379386	+	0.925238i	$0.376135\pi$
$864$	0	0
$865$	53.8999	1.83265
$866$	75.3084	2.55908
$867$	0	0
$868$	31.9941	1.08595
$869$	2.99897	0.101733
$870$	0	0
$871$	−70.4438	−2.38690
$872$	−0.384106	−0.0130075
$873$	0	0
$874$	−5.99906	−0.202921
$875$	13.0909	0.442552
$876$	0	0
$877$	−3.67076	−0.123953	−0.0619763	−	0.998078i	$-0.519740\pi$
$877$	−3.67076	−0.123953	−0.0619763	+	0.998078i	$0.519740\pi$
$878$	−40.7852	−1.37643
$879$	0	0
$880$	−2.93684	−0.0990008
$881$	6.66693	0.224615	0.112307	−	0.993674i	$-0.464176\pi$
$881$	6.66693	0.224615	0.112307	+	0.993674i	$0.464176\pi$
$882$	0	0
$883$	19.4496	0.654530	0.327265	−	0.944933i	$-0.393873\pi$
$883$	19.4496	0.654530	0.327265	+	0.944933i	$0.393873\pi$
$884$	−61.8749	−2.08108
$885$	0	0
$886$	59.5693	2.00127
$887$	6.36719	0.213789	0.106895	−	0.994270i	$-0.465909\pi$
$887$	6.36719	0.213789	0.106895	+	0.994270i	$0.465909\pi$
$888$	0	0
$889$	23.4078	0.785074
$890$	−107.459	−3.60205
$891$	0	0
$892$	−37.9833	−1.27177
$893$	6.75806	0.226150
$894$	0	0
$895$	1.88869	0.0631320
$896$	−29.3084	−0.979125
$897$	0	0
$898$	71.5754	2.38850
$899$	−5.79860	−0.193394
$900$	0	0
$901$	−32.0400	−1.06741
$902$	22.6470	0.754062
$903$	0	0
$904$	28.0779	0.933857
$905$	−58.7970	−1.95448
$906$	0	0
$907$	50.3211	1.67089	0.835443	−	0.549578i	$-0.185211\pi$
$907$	50.3211	1.67089	0.835443	+	0.549578i	$0.185211\pi$
$908$	−44.8257	−1.48759
$909$	0	0
$910$	50.1002	1.66081
$911$	−44.4615	−1.47307	−0.736537	−	0.676397i	$-0.763541\pi$
$911$	−44.4615	−1.47307	−0.736537	+	0.676397i	$0.763541\pi$
$912$	0	0
$913$	11.1523	0.369088
$914$	8.56308	0.283241
$915$	0	0
$916$	−34.6792	−1.14583
$917$	27.9489	0.922955
$918$	0	0
$919$	48.7767	1.60899	0.804497	−	0.593957i	$-0.202435\pi$
$919$	48.7767	1.60899	0.804497	+	0.593957i	$0.202435\pi$
$920$	−6.31985	−0.208359
$921$	0	0
$922$	−35.0179	−1.15325
$923$	37.1595	1.22312
$924$	0	0
$925$	−13.8398	−0.455051
$926$	71.9522	2.36449
$927$	0	0
$928$	6.55469	0.215168
$929$	−41.3850	−1.35780	−0.678898	−	0.734232i	$-0.737542\pi$
$929$	−41.3850	−1.35780	−0.678898	+	0.734232i	$0.737542\pi$
$930$	0	0
$931$	−9.89922	−0.324434
$932$	−32.0586	−1.05011
$933$	0	0
$934$	62.2433	2.03666
$935$	−15.6590	−0.512103
$936$	0	0
$937$	15.0228	0.490774	0.245387	−	0.969425i	$-0.421085\pi$
$937$	15.0228	0.490774	0.245387	+	0.969425i	$0.421085\pi$
$938$	−63.2578	−2.06544
$939$	0	0
$940$	20.8202	0.679079
$941$	28.3415	0.923907	0.461954	−	0.886904i	$-0.347149\pi$
$941$	28.3415	0.923907	0.461954	+	0.886904i	$0.347149\pi$
$942$	0	0
$943$	−7.82902	−0.254948
$944$	6.62537	0.215637
$945$	0	0
$946$	15.2839	0.496922
$947$	−14.4377	−0.469164	−0.234582	−	0.972096i	$-0.575372\pi$
$947$	−14.4377	−0.469164	−0.234582	+	0.972096i	$0.575372\pi$
$948$	0	0
$949$	−76.4864	−2.48285
$950$	−14.0260	−0.455064
$951$	0	0
$952$	−18.9997	−0.615784
$953$	13.5038	0.437431	0.218715	−	0.975789i	$-0.429813\pi$
$953$	13.5038	0.437431	0.218715	+	0.975789i	$0.429813\pi$
$954$	0	0
$955$	38.1764	1.23536
$956$	−11.4320	−0.369737
$957$	0	0
$958$	64.7648	2.09246
$959$	8.81736	0.284727
$960$	0	0
$961$	2.62375	0.0846371
$962$	60.3050	1.94431
$963$	0	0
$964$	−34.9469	−1.12556
$965$	30.2559	0.973972
$966$	0	0
$967$	25.4673	0.818974	0.409487	−	0.912316i	$-0.365708\pi$
$967$	25.4673	0.818974	0.409487	+	0.912316i	$0.365708\pi$
$968$	21.7892	0.700330
$969$	0	0
$970$	45.6566	1.46595
$971$	13.8575	0.444708	0.222354	−	0.974966i	$-0.428626\pi$
$971$	13.8575	0.444708	0.222354	+	0.974966i	$0.428626\pi$
$972$	0	0
$973$	−15.9181	−0.510311
$974$	29.3008	0.938858
$975$	0	0
$976$	−0.0162252	−0.000519355 0
$977$	−10.0587	−0.321805	−0.160902	−	0.986970i	$-0.551440\pi$
$977$	−10.0587	−0.321805	−0.160902	+	0.986970i	$0.551440\pi$
$978$	0	0
$979$	22.7714	0.727776
$980$	−30.4974	−0.974204
$981$	0	0
$982$	−37.5291	−1.19760
$983$	33.1846	1.05843	0.529213	−	0.848489i	$-0.322487\pi$
$983$	33.1846	1.05843	0.529213	+	0.848489i	$0.322487\pi$
$984$	0	0
$985$	32.0482	1.02114
$986$	10.0702	0.320701
$987$	0	0
$988$	36.8603	1.17268
$989$	−5.28362	−0.168009
$990$	0	0
$991$	−60.1166	−1.90967	−0.954834	−	0.297141i	$-0.903967\pi$
$991$	−60.1166	−1.90967	−0.954834	+	0.297141i	$0.903967\pi$
$992$	−38.0080	−1.20676
$993$	0	0
$994$	33.3688	1.05839
$995$	19.5966	0.621253
$996$	0	0
$997$	−6.86405	−0.217387	−0.108693	−	0.994075i	$-0.534667\pi$
$997$	−6.86405	−0.217387	−0.108693	+	0.994075i	$0.534667\pi$
$998$	−12.6720	−0.401126
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.t.1.3	✓	22
3.2	odd	2		6003.2.a.u.1.20	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.3	✓	22	1.1	even	1	trivial
6003.2.a.u.1.20	yes	22	3.2	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.24483 q^{2} +3.03928 q^{4} +2.70888 q^{5} -1.81542 q^{7} -2.33301 q^{8} -6.08099 q^{10} +1.28860 q^{11} +4.53826 q^{13} +4.07531 q^{14} -0.841340 q^{16} -4.48595 q^{17} +2.67238 q^{19} +8.23305 q^{20} -2.89270 q^{22} +1.00000 q^{23} +2.33804 q^{25} -10.1876 q^{26} -5.51756 q^{28} +1.00000 q^{29} -5.79860 q^{31} +6.55469 q^{32} +10.0702 q^{34} -4.91775 q^{35} -5.91943 q^{37} -5.99906 q^{38} -6.31985 q^{40} -7.82902 q^{41} -5.28362 q^{43} +3.91642 q^{44} -2.24483 q^{46} +2.52885 q^{47} -3.70427 q^{49} -5.24851 q^{50} +13.7930 q^{52} +7.14230 q^{53} +3.49067 q^{55} +4.23538 q^{56} -2.24483 q^{58} -7.87478 q^{59} +0.0192849 q^{61} +13.0169 q^{62} -13.0315 q^{64} +12.2936 q^{65} -15.5222 q^{67} -13.6340 q^{68} +11.0395 q^{70} +8.18804 q^{71} -16.8537 q^{73} +13.2881 q^{74} +8.12212 q^{76} -2.33935 q^{77} +2.32730 q^{79} -2.27909 q^{80} +17.5749 q^{82} +8.65460 q^{83} -12.1519 q^{85} +11.8608 q^{86} -3.00632 q^{88} +17.6714 q^{89} -8.23883 q^{91} +3.03928 q^{92} -5.67686 q^{94} +7.23917 q^{95} -7.50809 q^{97} +8.31546 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34}+ \cdots - 28 q^{98}+O(q^{100}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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