LMFDB - Embedded newform 1240.2.a.m.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1240,2,Mod(1,1240)] 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("1240.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1240 = 2^{3} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1240.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-1,0,6,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.90144985064$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 19x^{4} + 15x^{3} + 98x^{2} - 44x - 108 $ x^6 - x^5 - 19x^4 + 15x^3 + 98x^2 - 44x - 108
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.45950$ of defining polynomial
Character	$\chi$	$=$	1240.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.45950 q^{3} +1.00000 q^{5} -3.90059 q^{7} -0.869870 q^{9} +5.72206 q^{11} +4.22537 q^{13} -1.45950 q^{15} -5.18156 q^{17} -4.37318 q^{19} +5.69290 q^{21} -5.60498 q^{23} +1.00000 q^{25} +5.64806 q^{27} +6.34544 q^{29} +1.00000 q^{31} -8.35133 q^{33} -3.90059 q^{35} +10.6841 q^{37} -6.16691 q^{39} +9.97815 q^{41} +2.85678 q^{43} -0.869870 q^{45} -1.79231 q^{47} +8.21461 q^{49} +7.56247 q^{51} -4.34168 q^{53} +5.72206 q^{55} +6.38264 q^{57} -1.47485 q^{59} +9.11935 q^{61} +3.39301 q^{63} +4.22537 q^{65} -4.96583 q^{67} +8.18044 q^{69} +2.54582 q^{71} +15.5855 q^{73} -1.45950 q^{75} -22.3194 q^{77} +5.02728 q^{79} -5.63372 q^{81} +14.5221 q^{83} -5.18156 q^{85} -9.26115 q^{87} +15.0626 q^{89} -16.4814 q^{91} -1.45950 q^{93} -4.37318 q^{95} -8.30449 q^{97} -4.97745 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + 6 q^{5} - 2 q^{7} + 21 q^{9} + 8 q^{11} + 4 q^{13} - q^{15} + 3 q^{17} - 5 q^{19} + 8 q^{21} - 12 q^{23} + 6 q^{25} - 7 q^{27} - 4 q^{29} + 6 q^{31} + 14 q^{33} - 2 q^{35} + 3 q^{37} + 10 q^{39}+ \cdots - 40 q^{99}+O(q^{100}) $ 6 * q - q^3 + 6 * q^5 - 2 * q^7 + 21 * q^9 + 8 * q^11 + 4 * q^13 - q^15 + 3 * q^17 - 5 * q^19 + 8 * q^21 - 12 * q^23 + 6 * q^25 - 7 * q^27 - 4 * q^29 + 6 * q^31 + 14 * q^33 - 2 * q^35 + 3 * q^37 + 10 * q^39 + 17 * q^41 - 17 * q^43 + 21 * q^45 - 6 * q^47 + 34 * q^49 + 23 * q^51 + 9 * q^53 + 8 * q^55 - q^57 + 39 * q^59 + 2 * q^61 - 16 * q^63 + 4 * q^65 + 12 * q^67 - 8 * q^69 + 21 * q^71 + 43 * q^73 - q^75 + 20 * q^77 - 2 * q^79 + 50 * q^81 - 5 * q^83 + 3 * q^85 - 44 * q^87 + 6 * q^89 + 24 * q^91 - q^93 - 5 * q^95 - 12 * q^97 - 40 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.45950	−0.842641	−0.421320	−	0.906912i	$-0.638433\pi$
$3$	−1.45950	−0.842641	−0.421320	+	0.906912i	$0.638433\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.90059	−1.47428	−0.737142	−	0.675737i	$-0.763825\pi$
$7$	−3.90059	−1.47428	−0.737142	+	0.675737i	$0.763825\pi$
$8$	0	0
$9$	−0.869870	−0.289957
$10$	0	0
$11$	5.72206	1.72527	0.862633	−	0.505830i	$-0.168814\pi$
$11$	5.72206	1.72527	0.862633	+	0.505830i	$0.168814\pi$
$12$	0	0
$13$	4.22537	1.17191	0.585953	−	0.810345i	$-0.300720\pi$
$13$	4.22537	1.17191	0.585953	+	0.810345i	$0.300720\pi$
$14$	0	0
$15$	−1.45950	−0.376840
$16$	0	0
$17$	−5.18156	−1.25671	−0.628356	−	0.777926i	$-0.716272\pi$
$17$	−5.18156	−1.25671	−0.628356	+	0.777926i	$0.716272\pi$
$18$	0	0
$19$	−4.37318	−1.00328	−0.501638	−	0.865078i	$-0.667269\pi$
$19$	−4.37318	−1.00328	−0.501638	+	0.865078i	$0.667269\pi$
$20$	0	0
$21$	5.69290	1.24229
$22$	0	0
$23$	−5.60498	−1.16872	−0.584359	−	0.811495i	$-0.698654\pi$
$23$	−5.60498	−1.16872	−0.584359	+	0.811495i	$0.698654\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.64806	1.08697
$28$	0	0
$29$	6.34544	1.17832	0.589159	−	0.808017i	$-0.299459\pi$
$29$	6.34544	1.17832	0.589159	+	0.808017i	$0.299459\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−8.35133	−1.45378
$34$	0	0
$35$	−3.90059	−0.659320
$36$	0	0
$37$	10.6841	1.75646	0.878231	−	0.478237i	$-0.158724\pi$
$37$	10.6841	1.75646	0.878231	+	0.478237i	$0.158724\pi$
$38$	0	0
$39$	−6.16691	−0.987496
$40$	0	0
$41$	9.97815	1.55833	0.779163	−	0.626822i	$-0.215645\pi$
$41$	9.97815	1.55833	0.779163	+	0.626822i	$0.215645\pi$
$42$	0	0
$43$	2.85678	0.435655	0.217827	−	0.975987i	$-0.430103\pi$
$43$	2.85678	0.435655	0.217827	+	0.975987i	$0.430103\pi$
$44$	0	0
$45$	−0.869870	−0.129673
$46$	0	0
$47$	−1.79231	−0.261435	−0.130717	−	0.991420i	$-0.541728\pi$
$47$	−1.79231	−0.261435	−0.130717	+	0.991420i	$0.541728\pi$
$48$	0	0
$49$	8.21461	1.17352
$50$	0	0
$51$	7.56247	1.05896
$52$	0	0
$53$	−4.34168	−0.596376	−0.298188	−	0.954507i	$-0.596382\pi$
$53$	−4.34168	−0.596376	−0.298188	+	0.954507i	$0.596382\pi$
$54$	0	0
$55$	5.72206	0.771563
$56$	0	0
$57$	6.38264	0.845401
$58$	0	0
$59$	−1.47485	−0.192009	−0.0960043	−	0.995381i	$-0.530606\pi$
$59$	−1.47485	−0.192009	−0.0960043	+	0.995381i	$0.530606\pi$
$60$	0	0
$61$	9.11935	1.16761	0.583806	−	0.811893i	$-0.301563\pi$
$61$	9.11935	1.16761	0.583806	+	0.811893i	$0.301563\pi$
$62$	0	0
$63$	3.39301	0.427479
$64$	0	0
$65$	4.22537	0.524093
$66$	0	0
$67$	−4.96583	−0.606673	−0.303337	−	0.952883i	$-0.598101\pi$
$67$	−4.96583	−0.606673	−0.303337	+	0.952883i	$0.598101\pi$
$68$	0	0
$69$	8.18044	0.984810
$70$	0	0
$71$	2.54582	0.302133	0.151066	−	0.988524i	$-0.451729\pi$
$71$	2.54582	0.302133	0.151066	+	0.988524i	$0.451729\pi$
$72$	0	0
$73$	15.5855	1.82414	0.912070	−	0.410035i	$-0.134483\pi$
$73$	15.5855	1.82414	0.912070	+	0.410035i	$0.134483\pi$
$74$	0	0
$75$	−1.45950	−0.168528
$76$	0	0
$77$	−22.3194	−2.54353
$78$	0	0
$79$	5.02728	0.565613	0.282806	−	0.959177i	$-0.408735\pi$
$79$	5.02728	0.565613	0.282806	+	0.959177i	$0.408735\pi$
$80$	0	0
$81$	−5.63372	−0.625968
$82$	0	0
$83$	14.5221	1.59401	0.797005	−	0.603972i	$-0.206416\pi$
$83$	14.5221	1.59401	0.797005	+	0.603972i	$0.206416\pi$
$84$	0	0
$85$	−5.18156	−0.562019
$86$	0	0
$87$	−9.26115	−0.992899
$88$	0	0
$89$	15.0626	1.59664	0.798318	−	0.602236i	$-0.205724\pi$
$89$	15.0626	1.59664	0.798318	+	0.602236i	$0.205724\pi$
$90$	0	0
$91$	−16.4814	−1.72772
$92$	0	0
$93$	−1.45950	−0.151343
$94$	0	0
$95$	−4.37318	−0.448679
$96$	0	0
$97$	−8.30449	−0.843193	−0.421597	−	0.906784i	$-0.638530\pi$
$97$	−8.30449	−0.843193	−0.421597	+	0.906784i	$0.638530\pi$
$98$	0	0
$99$	−4.97745	−0.500253
$100$	0	0
$101$	−7.25537	−0.721936	−0.360968	−	0.932578i	$-0.617554\pi$
$101$	−7.25537	−0.721936	−0.360968	+	0.932578i	$0.617554\pi$
$102$	0	0
$103$	7.21461	0.710876	0.355438	−	0.934700i	$-0.384332\pi$
$103$	7.21461	0.710876	0.355438	+	0.934700i	$0.384332\pi$
$104$	0	0
$105$	5.69290	0.555570
$106$	0	0
$107$	12.3263	1.19163	0.595815	−	0.803122i	$-0.296829\pi$
$107$	12.3263	1.19163	0.595815	+	0.803122i	$0.296829\pi$
$108$	0	0
$109$	−9.36656	−0.897154	−0.448577	−	0.893744i	$-0.648069\pi$
$109$	−9.36656	−0.897154	−0.448577	+	0.893744i	$0.648069\pi$
$110$	0	0
$111$	−15.5935	−1.48007
$112$	0	0
$113$	−20.7144	−1.94865	−0.974326	−	0.225143i	$-0.927715\pi$
$113$	−20.7144	−1.94865	−0.974326	+	0.225143i	$0.927715\pi$
$114$	0	0
$115$	−5.60498	−0.522667
$116$	0	0
$117$	−3.67552	−0.339802
$118$	0	0
$119$	20.2111	1.85275
$120$	0	0
$121$	21.7420	1.97655
$122$	0	0
$123$	−14.5631	−1.31311
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.46914	0.130365	0.0651826	−	0.997873i	$-0.479237\pi$
$127$	1.46914	0.130365	0.0651826	+	0.997873i	$0.479237\pi$
$128$	0	0
$129$	−4.16946	−0.367101
$130$	0	0
$131$	14.7363	1.28752	0.643758	−	0.765229i	$-0.277374\pi$
$131$	14.7363	1.28752	0.643758	+	0.765229i	$0.277374\pi$
$132$	0	0
$133$	17.0580	1.47911
$134$	0	0
$135$	5.64806	0.486108
$136$	0	0
$137$	−7.28984	−0.622813	−0.311407	−	0.950277i	$-0.600800\pi$
$137$	−7.28984	−0.622813	−0.311407	+	0.950277i	$0.600800\pi$
$138$	0	0
$139$	13.6784	1.16018	0.580092	−	0.814551i	$-0.303017\pi$
$139$	13.6784	1.16018	0.580092	+	0.814551i	$0.303017\pi$
$140$	0	0
$141$	2.61587	0.220296
$142$	0	0
$143$	24.1778	2.02185
$144$	0	0
$145$	6.34544	0.526960
$146$	0	0
$147$	−11.9892	−0.988852
$148$	0	0
$149$	−3.73511	−0.305992	−0.152996	−	0.988227i	$-0.548892\pi$
$149$	−3.73511	−0.305992	−0.152996	+	0.988227i	$0.548892\pi$
$150$	0	0
$151$	−6.40390	−0.521142	−0.260571	−	0.965455i	$-0.583911\pi$
$151$	−6.40390	−0.521142	−0.260571	+	0.965455i	$0.583911\pi$
$152$	0	0
$153$	4.50728	0.364392
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−0.699513	−0.0558272	−0.0279136	−	0.999610i	$-0.508886\pi$
$157$	−0.699513	−0.0558272	−0.0279136	+	0.999610i	$0.508886\pi$
$158$	0	0
$159$	6.33667	0.502531
$160$	0	0
$161$	21.8627	1.72302
$162$	0	0
$163$	6.87530	0.538515	0.269257	−	0.963068i	$-0.413222\pi$
$163$	6.87530	0.538515	0.269257	+	0.963068i	$0.413222\pi$
$164$	0	0
$165$	−8.35133	−0.650150
$166$	0	0
$167$	−12.7877	−0.989542	−0.494771	−	0.869023i	$-0.664748\pi$
$167$	−12.7877	−0.989542	−0.494771	+	0.869023i	$0.664748\pi$
$168$	0	0
$169$	4.85375	0.373365
$170$	0	0
$171$	3.80410	0.290906
$172$	0	0
$173$	7.10167	0.539930	0.269965	−	0.962870i	$-0.412988\pi$
$173$	7.10167	0.539930	0.269965	+	0.962870i	$0.412988\pi$
$174$	0	0
$175$	−3.90059	−0.294857
$176$	0	0
$177$	2.15253	0.161794
$178$	0	0
$179$	−16.5390	−1.23619	−0.618093	−	0.786105i	$-0.712094\pi$
$179$	−16.5390	−1.23619	−0.618093	+	0.786105i	$0.712094\pi$
$180$	0	0
$181$	−1.83035	−0.136049	−0.0680243	−	0.997684i	$-0.521670\pi$
$181$	−1.83035	−0.136049	−0.0680243	+	0.997684i	$0.521670\pi$
$182$	0	0
$183$	−13.3097	−0.983878
$184$	0	0
$185$	10.6841	0.785514
$186$	0	0
$187$	−29.6492	−2.16816
$188$	0	0
$189$	−22.0308	−1.60250
$190$	0	0
$191$	−9.98279	−0.722329	−0.361165	−	0.932502i	$-0.617621\pi$
$191$	−9.98279	−0.722329	−0.361165	+	0.932502i	$0.617621\pi$
$192$	0	0
$193$	−2.19882	−0.158274	−0.0791372	−	0.996864i	$-0.525217\pi$
$193$	−2.19882	−0.158274	−0.0791372	+	0.996864i	$0.525217\pi$
$194$	0	0
$195$	−6.16691	−0.441622
$196$	0	0
$197$	23.2365	1.65553	0.827766	−	0.561073i	$-0.189611\pi$
$197$	23.2365	1.65553	0.827766	+	0.561073i	$0.189611\pi$
$198$	0	0
$199$	0.145087	0.0102850	0.00514249	−	0.999987i	$-0.498363\pi$
$199$	0.145087	0.0102850	0.00514249	+	0.999987i	$0.498363\pi$
$200$	0	0
$201$	7.24762	0.511208
$202$	0	0
$203$	−24.7510	−1.73718
$204$	0	0
$205$	9.97815	0.696904
$206$	0	0
$207$	4.87560	0.338878
$208$	0	0
$209$	−25.0236	−1.73092
$210$	0	0
$211$	−2.40681	−0.165692	−0.0828459	−	0.996562i	$-0.526401\pi$
$211$	−2.40681	−0.165692	−0.0828459	+	0.996562i	$0.526401\pi$
$212$	0	0
$213$	−3.71561	−0.254589
$214$	0	0
$215$	2.85678	0.194831
$216$	0	0
$217$	−3.90059	−0.264789
$218$	0	0
$219$	−22.7469	−1.53709
$220$	0	0
$221$	−21.8940	−1.47275
$222$	0	0
$223$	−7.25304	−0.485699	−0.242850	−	0.970064i	$-0.578082\pi$
$223$	−7.25304	−0.485699	−0.242850	+	0.970064i	$0.578082\pi$
$224$	0	0
$225$	−0.869870	−0.0579913
$226$	0	0
$227$	−8.77167	−0.582196	−0.291098	−	0.956693i	$-0.594021\pi$
$227$	−8.77167	−0.582196	−0.291098	+	0.956693i	$0.594021\pi$
$228$	0	0
$229$	−21.7896	−1.43990	−0.719948	−	0.694029i	$-0.755834\pi$
$229$	−21.7896	−1.43990	−0.719948	+	0.694029i	$0.755834\pi$
$230$	0	0
$231$	32.5751	2.14329
$232$	0	0
$233$	−17.4062	−1.14032	−0.570158	−	0.821535i	$-0.693118\pi$
$233$	−17.4062	−1.14032	−0.570158	+	0.821535i	$0.693118\pi$
$234$	0	0
$235$	−1.79231	−0.116917
$236$	0	0
$237$	−7.33729	−0.476608
$238$	0	0
$239$	13.1971	0.853649	0.426825	−	0.904334i	$-0.359632\pi$
$239$	13.1971	0.853649	0.426825	+	0.904334i	$0.359632\pi$
$240$	0	0
$241$	−18.9221	−1.21888	−0.609441	−	0.792831i	$-0.708606\pi$
$241$	−18.9221	−1.21888	−0.609441	+	0.792831i	$0.708606\pi$
$242$	0	0
$243$	−8.72180	−0.559503
$244$	0	0
$245$	8.21461	0.524812
$246$	0	0
$247$	−18.4783	−1.17575
$248$	0	0
$249$	−21.1950	−1.34318
$250$	0	0
$251$	25.5439	1.61232	0.806159	−	0.591700i	$-0.201543\pi$
$251$	25.5439	1.61232	0.806159	+	0.591700i	$0.201543\pi$
$252$	0	0
$253$	−32.0720	−2.01635
$254$	0	0
$255$	7.56247	0.473580
$256$	0	0
$257$	10.6013	0.661289	0.330644	−	0.943755i	$-0.392734\pi$
$257$	10.6013	0.661289	0.330644	+	0.943755i	$0.392734\pi$
$258$	0	0
$259$	−41.6745	−2.58953
$260$	0	0
$261$	−5.51971	−0.341661
$262$	0	0
$263$	17.5225	1.08048	0.540241	−	0.841510i	$-0.318333\pi$
$263$	17.5225	1.08048	0.540241	+	0.841510i	$0.318333\pi$
$264$	0	0
$265$	−4.34168	−0.266708
$266$	0	0
$267$	−21.9839	−1.34539
$268$	0	0
$269$	21.2207	1.29385	0.646926	−	0.762553i	$-0.276054\pi$
$269$	21.2207	1.29385	0.646926	+	0.762553i	$0.276054\pi$
$270$	0	0
$271$	−31.4518	−1.91056	−0.955281	−	0.295698i	$-0.904448\pi$
$271$	−31.4518	−1.91056	−0.955281	+	0.295698i	$0.904448\pi$
$272$	0	0
$273$	24.0546	1.45585
$274$	0	0
$275$	5.72206	0.345053
$276$	0	0
$277$	−25.4144	−1.52700	−0.763500	−	0.645808i	$-0.776521\pi$
$277$	−25.4144	−1.52700	−0.763500	+	0.645808i	$0.776521\pi$
$278$	0	0
$279$	−0.869870	−0.0520778
$280$	0	0
$281$	2.86761	0.171067	0.0855337	−	0.996335i	$-0.472740\pi$
$281$	2.86761	0.171067	0.0855337	+	0.996335i	$0.472740\pi$
$282$	0	0
$283$	15.1591	0.901117	0.450559	−	0.892747i	$-0.351225\pi$
$283$	15.1591	0.901117	0.450559	+	0.892747i	$0.351225\pi$
$284$	0	0
$285$	6.38264	0.378075
$286$	0	0
$287$	−38.9207	−2.29742
$288$	0	0
$289$	9.84855	0.579327
$290$	0	0
$291$	12.1204	0.710509
$292$	0	0
$293$	−18.4608	−1.07849	−0.539245	−	0.842149i	$-0.681290\pi$
$293$	−18.4608	−1.07849	−0.539245	+	0.842149i	$0.681290\pi$
$294$	0	0
$295$	−1.47485	−0.0858688
$296$	0	0
$297$	32.3186	1.87531
$298$	0	0
$299$	−23.6831	−1.36963
$300$	0	0
$301$	−11.1431	−0.642279
$302$	0	0
$303$	10.5892	0.608333
$304$	0	0
$305$	9.11935	0.522172
$306$	0	0
$307$	3.51482	0.200601	0.100301	−	0.994957i	$-0.468020\pi$
$307$	3.51482	0.200601	0.100301	+	0.994957i	$0.468020\pi$
$308$	0	0
$309$	−10.5297	−0.599013
$310$	0	0
$311$	24.2858	1.37712	0.688562	−	0.725177i	$-0.258242\pi$
$311$	24.2858	1.37712	0.688562	+	0.725177i	$0.258242\pi$
$312$	0	0
$313$	−0.340150	−0.0192264	−0.00961322	−	0.999954i	$-0.503060\pi$
$313$	−0.340150	−0.0192264	−0.00961322	+	0.999954i	$0.503060\pi$
$314$	0	0
$315$	3.39301	0.191174
$316$	0	0
$317$	31.6084	1.77531	0.887653	−	0.460513i	$-0.152335\pi$
$317$	31.6084	1.77531	0.887653	+	0.460513i	$0.152335\pi$
$318$	0	0
$319$	36.3090	2.03291
$320$	0	0
$321$	−17.9902	−1.00412
$322$	0	0
$323$	22.6599	1.26083
$324$	0	0
$325$	4.22537	0.234381
$326$	0	0
$327$	13.6705	0.755978
$328$	0	0
$329$	6.99106	0.385430
$330$	0	0
$331$	6.67495	0.366888	0.183444	−	0.983030i	$-0.441275\pi$
$331$	6.67495	0.366888	0.183444	+	0.983030i	$0.441275\pi$
$332$	0	0
$333$	−9.29381	−0.509298
$334$	0	0
$335$	−4.96583	−0.271313
$336$	0	0
$337$	−1.93160	−0.105221	−0.0526104	−	0.998615i	$-0.516754\pi$
$337$	−1.93160	−0.105221	−0.0526104	+	0.998615i	$0.516754\pi$
$338$	0	0
$339$	30.2327	1.64201
$340$	0	0
$341$	5.72206	0.309867
$342$	0	0
$343$	−4.73769	−0.255811
$344$	0	0
$345$	8.18044	0.440420
$346$	0	0
$347$	−23.2279	−1.24694	−0.623470	−	0.781847i	$-0.714278\pi$
$347$	−23.2279	−1.24694	−0.623470	+	0.781847i	$0.714278\pi$
$348$	0	0
$349$	−11.8449	−0.634040	−0.317020	−	0.948419i	$-0.602682\pi$
$349$	−11.8449	−0.634040	−0.317020	+	0.948419i	$0.602682\pi$
$350$	0	0
$351$	23.8651	1.27383
$352$	0	0
$353$	11.2350	0.597977	0.298989	−	0.954257i	$-0.403351\pi$
$353$	11.2350	0.597977	0.298989	+	0.954257i	$0.403351\pi$
$354$	0	0
$355$	2.54582	0.135118
$356$	0	0
$357$	−29.4981	−1.56120
$358$	0	0
$359$	−16.1672	−0.853272	−0.426636	−	0.904423i	$-0.640302\pi$
$359$	−16.1672	−0.853272	−0.426636	+	0.904423i	$0.640302\pi$
$360$	0	0
$361$	0.124679	0.00656204
$362$	0	0
$363$	−31.7324	−1.66552
$364$	0	0
$365$	15.5855	0.815780
$366$	0	0
$367$	−26.4315	−1.37971	−0.689857	−	0.723946i	$-0.742327\pi$
$367$	−26.4315	−1.37971	−0.689857	+	0.723946i	$0.742327\pi$
$368$	0	0
$369$	−8.67970	−0.451847
$370$	0	0
$371$	16.9351	0.879228
$372$	0	0
$373$	−20.8629	−1.08024	−0.540121	−	0.841587i	$-0.681621\pi$
$373$	−20.8629	−1.08024	−0.540121	+	0.841587i	$0.681621\pi$
$374$	0	0
$375$	−1.45950	−0.0753681
$376$	0	0
$377$	26.8118	1.38088
$378$	0	0
$379$	29.0204	1.49068	0.745340	−	0.666685i	$-0.232287\pi$
$379$	29.0204	1.49068	0.745340	+	0.666685i	$0.232287\pi$
$380$	0	0
$381$	−2.14421	−0.109851
$382$	0	0
$383$	−13.0342	−0.666018	−0.333009	−	0.942924i	$-0.608064\pi$
$383$	−13.0342	−0.666018	−0.333009	+	0.942924i	$0.608064\pi$
$384$	0	0
$385$	−22.3194	−1.13750
$386$	0	0
$387$	−2.48503	−0.126321
$388$	0	0
$389$	8.62491	0.437300	0.218650	−	0.975803i	$-0.429835\pi$
$389$	8.62491	0.437300	0.218650	+	0.975803i	$0.429835\pi$
$390$	0	0
$391$	29.0425	1.46874
$392$	0	0
$393$	−21.5076	−1.08491
$394$	0	0
$395$	5.02728	0.252950
$396$	0	0
$397$	−10.0354	−0.503660	−0.251830	−	0.967771i	$-0.581032\pi$
$397$	−10.0354	−0.503660	−0.251830	+	0.967771i	$0.581032\pi$
$398$	0	0
$399$	−24.8961	−1.24636
$400$	0	0
$401$	13.3605	0.667190	0.333595	−	0.942716i	$-0.391738\pi$
$401$	13.3605	0.667190	0.333595	+	0.942716i	$0.391738\pi$
$402$	0	0
$403$	4.22537	0.210481
$404$	0	0
$405$	−5.63372	−0.279942
$406$	0	0
$407$	61.1353	3.03037
$408$	0	0
$409$	−11.7984	−0.583391	−0.291696	−	0.956511i	$-0.594219\pi$
$409$	−11.7984	−0.583391	−0.291696	+	0.956511i	$0.594219\pi$
$410$	0	0
$411$	10.6395	0.524808
$412$	0	0
$413$	5.75277	0.283075
$414$	0	0
$415$	14.5221	0.712863
$416$	0	0
$417$	−19.9635	−0.977618
$418$	0	0
$419$	15.8581	0.774718	0.387359	−	0.921929i	$-0.373387\pi$
$419$	15.8581	0.774718	0.387359	+	0.921929i	$0.373387\pi$
$420$	0	0
$421$	1.98848	0.0969124	0.0484562	−	0.998825i	$-0.484570\pi$
$421$	1.98848	0.0969124	0.0484562	+	0.998825i	$0.484570\pi$
$422$	0	0
$423$	1.55907	0.0758048
$424$	0	0
$425$	−5.18156	−0.251343
$426$	0	0
$427$	−35.5708	−1.72139
$428$	0	0
$429$	−35.2875	−1.70369
$430$	0	0
$431$	−9.97418	−0.480439	−0.240220	−	0.970719i	$-0.577219\pi$
$431$	−9.97418	−0.480439	−0.240220	+	0.970719i	$0.577219\pi$
$432$	0	0
$433$	−0.0603717	−0.00290128	−0.00145064	−	0.999999i	$-0.500462\pi$
$433$	−0.0603717	−0.00290128	−0.00145064	+	0.999999i	$0.500462\pi$
$434$	0	0
$435$	−9.26115	−0.444038
$436$	0	0
$437$	24.5116	1.17255
$438$	0	0
$439$	2.08184	0.0993608	0.0496804	−	0.998765i	$-0.484180\pi$
$439$	2.08184	0.0993608	0.0496804	+	0.998765i	$0.484180\pi$
$440$	0	0
$441$	−7.14564	−0.340269
$442$	0	0
$443$	35.2416	1.67438	0.837190	−	0.546912i	$-0.184197\pi$
$443$	35.2416	1.67438	0.837190	+	0.546912i	$0.184197\pi$
$444$	0	0
$445$	15.0626	0.714037
$446$	0	0
$447$	5.45137	0.257841
$448$	0	0
$449$	2.35938	0.111346	0.0556731	−	0.998449i	$-0.482270\pi$
$449$	2.35938	0.111346	0.0556731	+	0.998449i	$0.482270\pi$
$450$	0	0
$451$	57.0956	2.68853
$452$	0	0
$453$	9.34647	0.439135
$454$	0	0
$455$	−16.4814	−0.772662
$456$	0	0
$457$	34.3574	1.60717	0.803585	−	0.595189i	$-0.202923\pi$
$457$	34.3574	1.60717	0.803585	+	0.595189i	$0.202923\pi$
$458$	0	0
$459$	−29.2658	−1.36601
$460$	0	0
$461$	−0.425939	−0.0198380	−0.00991898	−	0.999951i	$-0.503157\pi$
$461$	−0.425939	−0.0198380	−0.00991898	+	0.999951i	$0.503157\pi$
$462$	0	0
$463$	16.0543	0.746105	0.373052	−	0.927810i	$-0.378311\pi$
$463$	16.0543	0.746105	0.373052	+	0.927810i	$0.378311\pi$
$464$	0	0
$465$	−1.45950	−0.0676825
$466$	0	0
$467$	39.0818	1.80849	0.904246	−	0.427012i	$-0.140434\pi$
$467$	39.0818	1.80849	0.904246	+	0.427012i	$0.140434\pi$
$468$	0	0
$469$	19.3697	0.894409
$470$	0	0
$471$	1.02094	0.0470423
$472$	0	0
$473$	16.3467	0.751621
$474$	0	0
$475$	−4.37318	−0.200655
$476$	0	0
$477$	3.77670	0.172923
$478$	0	0
$479$	−5.83333	−0.266532	−0.133266	−	0.991080i	$-0.542546\pi$
$479$	−5.83333	−0.266532	−0.133266	+	0.991080i	$0.542546\pi$
$480$	0	0
$481$	45.1444	2.05841
$482$	0	0
$483$	−31.9086	−1.45189
$484$	0	0
$485$	−8.30449	−0.377087
$486$	0	0
$487$	−25.8420	−1.17101	−0.585507	−	0.810667i	$-0.699105\pi$
$487$	−25.8420	−1.17101	−0.585507	+	0.810667i	$0.699105\pi$
$488$	0	0
$489$	−10.0345	−0.453775
$490$	0	0
$491$	6.03834	0.272506	0.136253	−	0.990674i	$-0.456494\pi$
$491$	6.03834	0.272506	0.136253	+	0.990674i	$0.456494\pi$
$492$	0	0
$493$	−32.8793	−1.48081
$494$	0	0
$495$	−4.97745	−0.223720
$496$	0	0
$497$	−9.93019	−0.445430
$498$	0	0
$499$	−31.9441	−1.43002	−0.715008	−	0.699117i	$-0.753577\pi$
$499$	−31.9441	−1.43002	−0.715008	+	0.699117i	$0.753577\pi$
$500$	0	0
$501$	18.6636	0.833828
$502$	0	0
$503$	−4.01232	−0.178901	−0.0894503	−	0.995991i	$-0.528511\pi$
$503$	−4.01232	−0.178901	−0.0894503	+	0.995991i	$0.528511\pi$
$504$	0	0
$505$	−7.25537	−0.322860
$506$	0	0
$507$	−7.08403	−0.314613
$508$	0	0
$509$	1.57091	0.0696295	0.0348148	−	0.999394i	$-0.488916\pi$
$509$	1.57091	0.0696295	0.0348148	+	0.999394i	$0.488916\pi$
$510$	0	0
$511$	−60.7925	−2.68930
$512$	0	0
$513$	−24.7000	−1.09053
$514$	0	0
$515$	7.21461	0.317914
$516$	0	0
$517$	−10.2557	−0.451045
$518$	0	0
$519$	−10.3649	−0.454967
$520$	0	0
$521$	−30.6664	−1.34352	−0.671760	−	0.740769i	$-0.734461\pi$
$521$	−30.6664	−1.34352	−0.671760	+	0.740769i	$0.734461\pi$
$522$	0	0
$523$	11.8373	0.517609	0.258805	−	0.965930i	$-0.416671\pi$
$523$	11.8373	0.517609	0.258805	+	0.965930i	$0.416671\pi$
$524$	0	0
$525$	5.69290	0.248458
$526$	0	0
$527$	−5.18156	−0.225712
$528$	0	0
$529$	8.41575	0.365902
$530$	0	0
$531$	1.28292	0.0556742
$532$	0	0
$533$	42.1614	1.82621
$534$	0	0
$535$	12.3263	0.532913
$536$	0	0
$537$	24.1387	1.04166
$538$	0	0
$539$	47.0045	2.02463
$540$	0	0
$541$	−5.22523	−0.224650	−0.112325	−	0.993672i	$-0.535830\pi$
$541$	−5.22523	−0.224650	−0.112325	+	0.993672i	$0.535830\pi$
$542$	0	0
$543$	2.67138	0.114640
$544$	0	0
$545$	−9.36656	−0.401219
$546$	0	0
$547$	−29.1387	−1.24588	−0.622941	−	0.782269i	$-0.714062\pi$
$547$	−29.1387	−1.24588	−0.622941	+	0.782269i	$0.714062\pi$
$548$	0	0
$549$	−7.93265	−0.338557
$550$	0	0
$551$	−27.7497	−1.18218
$552$	0	0
$553$	−19.6093	−0.833874
$554$	0	0
$555$	−15.5935	−0.661906
$556$	0	0
$557$	−28.1486	−1.19269	−0.596347	−	0.802726i	$-0.703382\pi$
$557$	−28.1486	−1.19269	−0.596347	+	0.802726i	$0.703382\pi$
$558$	0	0
$559$	12.0710	0.510547
$560$	0	0
$561$	43.2729	1.82698
$562$	0	0
$563$	−18.1614	−0.765411	−0.382706	−	0.923870i	$-0.625008\pi$
$563$	−18.1614	−0.765411	−0.382706	+	0.923870i	$0.625008\pi$
$564$	0	0
$565$	−20.7144	−0.871463
$566$	0	0
$567$	21.9748	0.922856
$568$	0	0
$569$	16.7909	0.703910	0.351955	−	0.936017i	$-0.385517\pi$
$569$	16.7909	0.703910	0.351955	+	0.936017i	$0.385517\pi$
$570$	0	0
$571$	47.3293	1.98067	0.990335	−	0.138695i	$-0.0442909\pi$
$571$	47.3293	1.98067	0.990335	+	0.138695i	$0.0442909\pi$
$572$	0	0
$573$	14.5698	0.608664
$574$	0	0
$575$	−5.60498	−0.233744
$576$	0	0
$577$	34.9426	1.45468	0.727339	−	0.686278i	$-0.240757\pi$
$577$	34.9426	1.45468	0.727339	+	0.686278i	$0.240757\pi$
$578$	0	0
$579$	3.20917	0.133368
$580$	0	0
$581$	−56.6449	−2.35003
$582$	0	0
$583$	−24.8434	−1.02891
$584$	0	0
$585$	−3.67552	−0.151964
$586$	0	0
$587$	32.2345	1.33046	0.665231	−	0.746638i	$-0.268333\pi$
$587$	32.2345	1.33046	0.665231	+	0.746638i	$0.268333\pi$
$588$	0	0
$589$	−4.37318	−0.180194
$590$	0	0
$591$	−33.9136	−1.39502
$592$	0	0
$593$	−14.2700	−0.585997	−0.292999	−	0.956113i	$-0.594653\pi$
$593$	−14.2700	−0.585997	−0.292999	+	0.956113i	$0.594653\pi$
$594$	0	0
$595$	20.2111	0.828576
$596$	0	0
$597$	−0.211755	−0.00866654
$598$	0	0
$599$	−0.142475	−0.00582136	−0.00291068	−	0.999996i	$-0.500926\pi$
$599$	−0.142475	−0.00582136	−0.00291068	+	0.999996i	$0.500926\pi$
$600$	0	0
$601$	42.9575	1.75227	0.876136	−	0.482063i	$-0.160112\pi$
$601$	42.9575	1.75227	0.876136	+	0.482063i	$0.160112\pi$
$602$	0	0
$603$	4.31963	0.175909
$604$	0	0
$605$	21.7420	0.883938
$606$	0	0
$607$	42.8115	1.73767	0.868834	−	0.495104i	$-0.164870\pi$
$607$	42.8115	1.73767	0.868834	+	0.495104i	$0.164870\pi$
$608$	0	0
$609$	36.1239	1.46382
$610$	0	0
$611$	−7.57316	−0.306377
$612$	0	0
$613$	−5.70516	−0.230429	−0.115215	−	0.993341i	$-0.536756\pi$
$613$	−5.70516	−0.230429	−0.115215	+	0.993341i	$0.536756\pi$
$614$	0	0
$615$	−14.5631	−0.587240
$616$	0	0
$617$	33.0057	1.32876	0.664380	−	0.747395i	$-0.268696\pi$
$617$	33.0057	1.32876	0.664380	+	0.747395i	$0.268696\pi$
$618$	0	0
$619$	42.7016	1.71632	0.858161	−	0.513381i	$-0.171607\pi$
$619$	42.7016	1.71632	0.858161	+	0.513381i	$0.171607\pi$
$620$	0	0
$621$	−31.6572	−1.27036
$622$	0	0
$623$	−58.7532	−2.35390
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	36.5218	1.45854
$628$	0	0
$629$	−55.3605	−2.20737
$630$	0	0
$631$	−47.9692	−1.90962	−0.954812	−	0.297210i	$-0.903944\pi$
$631$	−47.9692	−1.90962	−0.954812	+	0.297210i	$0.903944\pi$
$632$	0	0
$633$	3.51273	0.139619
$634$	0	0
$635$	1.46914	0.0583011
$636$	0	0
$637$	34.7098	1.37525
$638$	0	0
$639$	−2.21453	−0.0876054
$640$	0	0
$641$	43.1333	1.70366	0.851831	−	0.523817i	$-0.175492\pi$
$641$	43.1333	1.70366	0.851831	+	0.523817i	$0.175492\pi$
$642$	0	0
$643$	28.8306	1.13697	0.568483	−	0.822695i	$-0.307530\pi$
$643$	28.8306	1.13697	0.568483	+	0.822695i	$0.307530\pi$
$644$	0	0
$645$	−4.16946	−0.164172
$646$	0	0
$647$	9.42108	0.370381	0.185190	−	0.982703i	$-0.440710\pi$
$647$	9.42108	0.370381	0.185190	+	0.982703i	$0.440710\pi$
$648$	0	0
$649$	−8.43916	−0.331266
$650$	0	0
$651$	5.69290	0.223122
$652$	0	0
$653$	−7.67141	−0.300205	−0.150103	−	0.988670i	$-0.547960\pi$
$653$	−7.67141	−0.300205	−0.150103	+	0.988670i	$0.547960\pi$
$654$	0	0
$655$	14.7363	0.575795
$656$	0	0
$657$	−13.5573	−0.528921
$658$	0	0
$659$	2.25020	0.0876554	0.0438277	−	0.999039i	$-0.486045\pi$
$659$	2.25020	0.0876554	0.0438277	+	0.999039i	$0.486045\pi$
$660$	0	0
$661$	−38.9574	−1.51527	−0.757633	−	0.652681i	$-0.773644\pi$
$661$	−38.9574	−1.51527	−0.757633	+	0.652681i	$0.773644\pi$
$662$	0	0
$663$	31.9542	1.24100
$664$	0	0
$665$	17.0580	0.661480
$666$	0	0
$667$	−35.5660	−1.37712
$668$	0	0
$669$	10.5858	0.409270
$670$	0	0
$671$	52.1815	2.01444
$672$	0	0
$673$	−29.7727	−1.14765	−0.573827	−	0.818976i	$-0.694542\pi$
$673$	−29.7727	−1.14765	−0.573827	+	0.818976i	$0.694542\pi$
$674$	0	0
$675$	5.64806	0.217394
$676$	0	0
$677$	−29.5784	−1.13679	−0.568394	−	0.822756i	$-0.692435\pi$
$677$	−29.5784	−1.13679	−0.568394	+	0.822756i	$0.692435\pi$
$678$	0	0
$679$	32.3924	1.24311
$680$	0	0
$681$	12.8022	0.490582
$682$	0	0
$683$	23.4165	0.896007	0.448003	−	0.894032i	$-0.352135\pi$
$683$	23.4165	0.896007	0.448003	+	0.894032i	$0.352135\pi$
$684$	0	0
$685$	−7.28984	−0.278531
$686$	0	0
$687$	31.8018	1.21331
$688$	0	0
$689$	−18.3452	−0.698897
$690$	0	0
$691$	−5.07647	−0.193118	−0.0965589	−	0.995327i	$-0.530784\pi$
$691$	−5.07647	−0.193118	−0.0965589	+	0.995327i	$0.530784\pi$
$692$	0	0
$693$	19.4150	0.737515
$694$	0	0
$695$	13.6784	0.518850
$696$	0	0
$697$	−51.7024	−1.95837
$698$	0	0
$699$	25.4042	0.960876
$700$	0	0
$701$	−23.7506	−0.897048	−0.448524	−	0.893771i	$-0.648050\pi$
$701$	−23.7506	−0.897048	−0.448524	+	0.893771i	$0.648050\pi$
$702$	0	0
$703$	−46.7236	−1.76222
$704$	0	0
$705$	2.61587	0.0985192
$706$	0	0
$707$	28.3002	1.06434
$708$	0	0
$709$	9.37961	0.352258	0.176129	−	0.984367i	$-0.443642\pi$
$709$	9.37961	0.352258	0.176129	+	0.984367i	$0.443642\pi$
$710$	0	0
$711$	−4.37308	−0.164003
$712$	0	0
$713$	−5.60498	−0.209908
$714$	0	0
$715$	24.1778	0.904200
$716$	0	0
$717$	−19.2611	−0.719319
$718$	0	0
$719$	−11.9138	−0.444309	−0.222155	−	0.975011i	$-0.571309\pi$
$719$	−11.9138	−0.444309	−0.222155	+	0.975011i	$0.571309\pi$
$720$	0	0
$721$	−28.1412	−1.04803
$722$	0	0
$723$	27.6168	1.02708
$724$	0	0
$725$	6.34544	0.235664
$726$	0	0
$727$	24.3586	0.903409	0.451704	−	0.892168i	$-0.350816\pi$
$727$	24.3586	0.903409	0.451704	+	0.892168i	$0.350816\pi$
$728$	0	0
$729$	29.6306	1.09743
$730$	0	0
$731$	−14.8026	−0.547493
$732$	0	0
$733$	6.66097	0.246028	0.123014	−	0.992405i	$-0.460744\pi$
$733$	6.66097	0.246028	0.123014	+	0.992405i	$0.460744\pi$
$734$	0	0
$735$	−11.9892	−0.442228
$736$	0	0
$737$	−28.4148	−1.04667
$738$	0	0
$739$	−12.0485	−0.443210	−0.221605	−	0.975137i	$-0.571129\pi$
$739$	−12.0485	−0.443210	−0.221605	+	0.975137i	$0.571129\pi$
$740$	0	0
$741$	26.9690	0.990731
$742$	0	0
$743$	27.5768	1.01170	0.505848	−	0.862623i	$-0.331180\pi$
$743$	27.5768	1.01170	0.505848	+	0.862623i	$0.331180\pi$
$744$	0	0
$745$	−3.73511	−0.136844
$746$	0	0
$747$	−12.6324	−0.462194
$748$	0	0
$749$	−48.0799	−1.75680
$750$	0	0
$751$	−22.2453	−0.811744	−0.405872	−	0.913930i	$-0.633032\pi$
$751$	−22.2453	−0.811744	−0.405872	+	0.913930i	$0.633032\pi$
$752$	0	0
$753$	−37.2812	−1.35860
$754$	0	0
$755$	−6.40390	−0.233062
$756$	0	0
$757$	8.51287	0.309406	0.154703	−	0.987961i	$-0.450558\pi$
$757$	8.51287	0.309406	0.154703	+	0.987961i	$0.450558\pi$
$758$	0	0
$759$	46.8090	1.69906
$760$	0	0
$761$	−24.5737	−0.890797	−0.445398	−	0.895333i	$-0.646938\pi$
$761$	−24.5737	−0.890797	−0.445398	+	0.895333i	$0.646938\pi$
$762$	0	0
$763$	36.5351	1.32266
$764$	0	0
$765$	4.50728	0.162961
$766$	0	0
$767$	−6.23177	−0.225016
$768$	0	0
$769$	7.88670	0.284402	0.142201	−	0.989838i	$-0.454582\pi$
$769$	7.88670	0.284402	0.142201	+	0.989838i	$0.454582\pi$
$770$	0	0
$771$	−15.4725	−0.557229
$772$	0	0
$773$	51.1007	1.83796	0.918982	−	0.394300i	$-0.129013\pi$
$773$	51.1007	1.83796	0.918982	+	0.394300i	$0.129013\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	60.8237	2.18204
$778$	0	0
$779$	−43.6362	−1.56343
$780$	0	0
$781$	14.5673	0.521260
$782$	0	0
$783$	35.8394	1.28080
$784$	0	0
$785$	−0.699513	−0.0249667
$786$	0	0
$787$	4.28458	0.152729	0.0763644	−	0.997080i	$-0.475669\pi$
$787$	4.28458	0.152729	0.0763644	+	0.997080i	$0.475669\pi$
$788$	0	0
$789$	−25.5740	−0.910458
$790$	0	0
$791$	80.7986	2.87287
$792$	0	0
$793$	38.5326	1.36833
$794$	0	0
$795$	6.33667	0.224739
$796$	0	0
$797$	13.4291	0.475682	0.237841	−	0.971304i	$-0.423560\pi$
$797$	13.4291	0.475682	0.237841	+	0.971304i	$0.423560\pi$
$798$	0	0
$799$	9.28695	0.328549
$800$	0	0
$801$	−13.1025	−0.462955
$802$	0	0
$803$	89.1810	3.14713
$804$	0	0
$805$	21.8627	0.770560
$806$	0	0
$807$	−30.9716	−1.09025
$808$	0	0
$809$	−41.4364	−1.45683	−0.728413	−	0.685139i	$-0.759742\pi$
$809$	−41.4364	−1.45683	−0.728413	+	0.685139i	$0.759742\pi$
$810$	0	0
$811$	4.71186	0.165456	0.0827278	−	0.996572i	$-0.473637\pi$
$811$	4.71186	0.165456	0.0827278	+	0.996572i	$0.473637\pi$
$812$	0	0
$813$	45.9038	1.60992
$814$	0	0
$815$	6.87530	0.240831
$816$	0	0
$817$	−12.4932	−0.437082
$818$	0	0
$819$	14.3367	0.500965
$820$	0	0
$821$	−48.3963	−1.68904	−0.844521	−	0.535522i	$-0.820115\pi$
$821$	−48.3963	−1.68904	−0.844521	+	0.535522i	$0.820115\pi$
$822$	0	0
$823$	0.436669	0.0152213	0.00761066	−	0.999971i	$-0.497577\pi$
$823$	0.436669	0.0152213	0.00761066	+	0.999971i	$0.497577\pi$
$824$	0	0
$825$	−8.35133	−0.290756
$826$	0	0
$827$	−37.3107	−1.29742	−0.648710	−	0.761036i	$-0.724691\pi$
$827$	−37.3107	−1.29742	−0.648710	+	0.761036i	$0.724691\pi$
$828$	0	0
$829$	−19.3190	−0.670977	−0.335488	−	0.942044i	$-0.608901\pi$
$829$	−19.3190	−0.670977	−0.335488	+	0.942044i	$0.608901\pi$
$830$	0	0
$831$	37.0922	1.28671
$832$	0	0
$833$	−42.5645	−1.47477
$834$	0	0
$835$	−12.7877	−0.442537
$836$	0	0
$837$	5.64806	0.195226
$838$	0	0
$839$	6.35098	0.219260	0.109630	−	0.993972i	$-0.465033\pi$
$839$	6.35098	0.219260	0.109630	+	0.993972i	$0.465033\pi$
$840$	0	0
$841$	11.2646	0.388435
$842$	0	0
$843$	−4.18527	−0.144148
$844$	0	0
$845$	4.85375	0.166974
$846$	0	0
$847$	−84.8066	−2.91399
$848$	0	0
$849$	−22.1247	−0.759318
$850$	0	0
$851$	−59.8844	−2.05281
$852$	0	0
$853$	1.34733	0.0461317	0.0230658	−	0.999734i	$-0.492657\pi$
$853$	1.34733	0.0461317	0.0230658	+	0.999734i	$0.492657\pi$
$854$	0	0
$855$	3.80410	0.130097
$856$	0	0
$857$	−27.4626	−0.938103	−0.469051	−	0.883171i	$-0.655404\pi$
$857$	−27.4626	−0.938103	−0.469051	+	0.883171i	$0.655404\pi$
$858$	0	0
$859$	26.8907	0.917500	0.458750	−	0.888565i	$-0.348297\pi$
$859$	26.8907	0.917500	0.458750	+	0.888565i	$0.348297\pi$
$860$	0	0
$861$	56.8046	1.93590
$862$	0	0
$863$	−7.71135	−0.262497	−0.131249	−	0.991349i	$-0.541899\pi$
$863$	−7.71135	−0.262497	−0.131249	+	0.991349i	$0.541899\pi$
$864$	0	0
$865$	7.10167	0.241464
$866$	0	0
$867$	−14.3739	−0.488164
$868$	0	0
$869$	28.7664	0.975833
$870$	0	0
$871$	−20.9825	−0.710964
$872$	0	0
$873$	7.22383	0.244489
$874$	0	0
$875$	−3.90059	−0.131864
$876$	0	0
$877$	19.2307	0.649375	0.324687	−	0.945821i	$-0.394741\pi$
$877$	19.2307	0.649375	0.324687	+	0.945821i	$0.394741\pi$
$878$	0	0
$879$	26.9434	0.908780
$880$	0	0
$881$	40.5857	1.36737	0.683683	−	0.729779i	$-0.260377\pi$
$881$	40.5857	1.36737	0.683683	+	0.729779i	$0.260377\pi$
$882$	0	0
$883$	−46.2849	−1.55761	−0.778805	−	0.627266i	$-0.784174\pi$
$883$	−46.2849	−1.55761	−0.778805	+	0.627266i	$0.784174\pi$
$884$	0	0
$885$	2.15253	0.0723566
$886$	0	0
$887$	23.7294	0.796754	0.398377	−	0.917222i	$-0.369574\pi$
$887$	23.7294	0.796754	0.398377	+	0.917222i	$0.369574\pi$
$888$	0	0
$889$	−5.73052	−0.192195
$890$	0	0
$891$	−32.2365	−1.07996
$892$	0	0
$893$	7.83808	0.262291
$894$	0	0
$895$	−16.5390	−0.552839
$896$	0	0
$897$	34.5654	1.15410
$898$	0	0
$899$	6.34544	0.211632
$900$	0	0
$901$	22.4967	0.749474
$902$	0	0
$903$	16.2634	0.541211
$904$	0	0
$905$	−1.83035	−0.0608427
$906$	0	0
$907$	−31.2559	−1.03784	−0.518918	−	0.854824i	$-0.673665\pi$
$907$	−31.2559	−1.03784	−0.518918	+	0.854824i	$0.673665\pi$
$908$	0	0
$909$	6.31122	0.209330
$910$	0	0
$911$	−14.6943	−0.486843	−0.243422	−	0.969921i	$-0.578270\pi$
$911$	−14.6943	−0.486843	−0.243422	+	0.969921i	$0.578270\pi$
$912$	0	0
$913$	83.0965	2.75009
$914$	0	0
$915$	−13.3097	−0.440004
$916$	0	0
$917$	−57.4803	−1.89817
$918$	0	0
$919$	−35.8610	−1.18294	−0.591472	−	0.806326i	$-0.701453\pi$
$919$	−35.8610	−1.18294	−0.591472	+	0.806326i	$0.701453\pi$
$920$	0	0
$921$	−5.12987	−0.169035
$922$	0	0
$923$	10.7570	0.354071
$924$	0	0
$925$	10.6841	0.351292
$926$	0	0
$927$	−6.27577	−0.206123
$928$	0	0
$929$	−0.627663	−0.0205930	−0.0102965	−	0.999947i	$-0.503278\pi$
$929$	−0.627663	−0.0205930	−0.0102965	+	0.999947i	$0.503278\pi$
$930$	0	0
$931$	−35.9239	−1.17736
$932$	0	0
$933$	−35.4451	−1.16042
$934$	0	0
$935$	−29.6492	−0.969633
$936$	0	0
$937$	3.67455	0.120042	0.0600211	−	0.998197i	$-0.480883\pi$
$937$	3.67455	0.120042	0.0600211	+	0.998197i	$0.480883\pi$
$938$	0	0
$939$	0.496448	0.0162010
$940$	0	0
$941$	34.1389	1.11290	0.556448	−	0.830883i	$-0.312164\pi$
$941$	34.1389	1.11290	0.556448	+	0.830883i	$0.312164\pi$
$942$	0	0
$943$	−55.9273	−1.82124
$944$	0	0
$945$	−22.0308	−0.716661
$946$	0	0
$947$	−33.6169	−1.09240	−0.546201	−	0.837654i	$-0.683927\pi$
$947$	−33.6169	−1.09240	−0.546201	+	0.837654i	$0.683927\pi$
$948$	0	0
$949$	65.8543	2.13772
$950$	0	0
$951$	−46.1324	−1.49595
$952$	0	0
$953$	9.45880	0.306401	0.153200	−	0.988195i	$-0.451042\pi$
$953$	9.45880	0.306401	0.153200	+	0.988195i	$0.451042\pi$
$954$	0	0
$955$	−9.98279	−0.323035
$956$	0	0
$957$	−52.9929	−1.71302
$958$	0	0
$959$	28.4347	0.918204
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−10.7223	−0.345521
$964$	0	0
$965$	−2.19882	−0.0707825
$966$	0	0
$967$	−5.27032	−0.169482	−0.0847411	−	0.996403i	$-0.527006\pi$
$967$	−5.27032	−0.169482	−0.0847411	+	0.996403i	$0.527006\pi$
$968$	0	0
$969$	−33.0720	−1.06243
$970$	0	0
$971$	1.41652	0.0454582	0.0227291	−	0.999742i	$-0.492764\pi$
$971$	1.41652	0.0454582	0.0227291	+	0.999742i	$0.492764\pi$
$972$	0	0
$973$	−53.3537	−1.71044
$974$	0	0
$975$	−6.16691	−0.197499
$976$	0	0
$977$	39.2413	1.25544	0.627721	−	0.778439i	$-0.283988\pi$
$977$	39.2413	1.25544	0.627721	+	0.778439i	$0.283988\pi$
$978$	0	0
$979$	86.1893	2.75462
$980$	0	0
$981$	8.14769	0.260136
$982$	0	0
$983$	−22.5884	−0.720457	−0.360228	−	0.932864i	$-0.617301\pi$
$983$	−22.5884	−0.720457	−0.360228	+	0.932864i	$0.617301\pi$
$984$	0	0
$985$	23.2365	0.740377
$986$	0	0
$987$	−10.2034	−0.324779
$988$	0	0
$989$	−16.0122	−0.509158
$990$	0	0
$991$	23.3804	0.742702	0.371351	−	0.928493i	$-0.378895\pi$
$991$	23.3804	0.742702	0.371351	+	0.928493i	$0.378895\pi$
$992$	0	0
$993$	−9.74206	−0.309155
$994$	0	0
$995$	0.145087	0.00459958
$996$	0	0
$997$	50.1014	1.58673	0.793363	−	0.608749i	$-0.208328\pi$
$997$	50.1014	1.58673	0.793363	+	0.608749i	$0.208328\pi$
$998$	0	0
$999$	60.3447	1.90922

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1240.2.a.m.1.3	✓	6
4.3	odd	2		2480.2.a.ba.1.4		6
5.4	even	2		6200.2.a.w.1.4		6
8.3	odd	2		9920.2.a.co.1.3		6
8.5	even	2		9920.2.a.cp.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1240.2.a.m.1.3	✓	6	1.1	even	1	trivial
2480.2.a.ba.1.4		6	4.3	odd	2
6200.2.a.w.1.4		6	5.4	even	2
9920.2.a.co.1.3		6	8.3	odd	2
9920.2.a.cp.1.4		6	8.5	even	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:	\( N \)	\(=\)	\( 1240 = 2^{3} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1240.a (trivial)

\(f(q)\)	\(=\)	\(q-1.45950 q^{3} +1.00000 q^{5} -3.90059 q^{7} -0.869870 q^{9} +5.72206 q^{11} +4.22537 q^{13} -1.45950 q^{15} -5.18156 q^{17} -4.37318 q^{19} +5.69290 q^{21} -5.60498 q^{23} +1.00000 q^{25} +5.64806 q^{27} +6.34544 q^{29} +1.00000 q^{31} -8.35133 q^{33} -3.90059 q^{35} +10.6841 q^{37} -6.16691 q^{39} +9.97815 q^{41} +2.85678 q^{43} -0.869870 q^{45} -1.79231 q^{47} +8.21461 q^{49} +7.56247 q^{51} -4.34168 q^{53} +5.72206 q^{55} +6.38264 q^{57} -1.47485 q^{59} +9.11935 q^{61} +3.39301 q^{63} +4.22537 q^{65} -4.96583 q^{67} +8.18044 q^{69} +2.54582 q^{71} +15.5855 q^{73} -1.45950 q^{75} -22.3194 q^{77} +5.02728 q^{79} -5.63372 q^{81} +14.5221 q^{83} -5.18156 q^{85} -9.26115 q^{87} +15.0626 q^{89} -16.4814 q^{91} -1.45950 q^{93} -4.37318 q^{95} -8.30449 q^{97} -4.97745 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + 6 q^{5} - 2 q^{7} + 21 q^{9} + 8 q^{11} + 4 q^{13} - q^{15} + 3 q^{17} - 5 q^{19} + 8 q^{21} - 12 q^{23} + 6 q^{25} - 7 q^{27} - 4 q^{29} + 6 q^{31} + 14 q^{33} - 2 q^{35} + 3 q^{37} + 10 q^{39}+ \cdots - 40 q^{99}+O(q^{100}) \) 6 * q - q^3 + 6 * q^5 - 2 * q^7 + 21 * q^9 + 8 * q^11 + 4 * q^13 - q^15 + 3 * q^17 - 5 * q^19 + 8 * q^21 - 12 * q^23 + 6 * q^25 - 7 * q^27 - 4 * q^29 + 6 * q^31 + 14 * q^33 - 2 * q^35 + 3 * q^37 + 10 * q^39 + 17 * q^41 - 17 * q^43 + 21 * q^45 - 6 * q^47 + 34 * q^49 + 23 * q^51 + 9 * q^53 + 8 * q^55 - q^57 + 39 * q^59 + 2 * q^61 - 16 * q^63 + 4 * q^65 + 12 * q^67 - 8 * q^69 + 21 * q^71 + 43 * q^73 - q^75 + 20 * q^77 - 2 * q^79 + 50 * q^81 - 5 * q^83 + 3 * q^85 - 44 * q^87 + 6 * q^89 + 24 * q^91 - q^93 - 5 * q^95 - 12 * q^97 - 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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