LMFDB - Embedded newform 8041.2.a.e.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8041,2,Mod(1,8041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8041, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8041 = 11 \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8041.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$64.2077082653$
Analytic rank:	$0$
Dimension:	$66$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.70598 q^{2} +0.804154 q^{3} +0.910358 q^{4} -1.55515 q^{5} -1.37187 q^{6} +1.24699 q^{7} +1.85890 q^{8} -2.35334 q^{9} +O(q^{10})$	\(q-1.70598 q^{2} +0.804154 q^{3} +0.910358 q^{4} -1.55515 q^{5} -1.37187 q^{6} +1.24699 q^{7} +1.85890 q^{8} -2.35334 q^{9} +2.65305 q^{10} -1.00000 q^{11} +0.732068 q^{12} +1.63357 q^{13} -2.12734 q^{14} -1.25058 q^{15} -4.99196 q^{16} -1.00000 q^{17} +4.01474 q^{18} +1.25046 q^{19} -1.41574 q^{20} +1.00277 q^{21} +1.70598 q^{22} -1.43802 q^{23} +1.49485 q^{24} -2.58152 q^{25} -2.78683 q^{26} -4.30491 q^{27} +1.13521 q^{28} +5.74338 q^{29} +2.13346 q^{30} +2.64533 q^{31} +4.79837 q^{32} -0.804154 q^{33} +1.70598 q^{34} -1.93925 q^{35} -2.14238 q^{36} +6.09996 q^{37} -2.13326 q^{38} +1.31364 q^{39} -2.89087 q^{40} -7.54817 q^{41} -1.71071 q^{42} +1.00000 q^{43} -0.910358 q^{44} +3.65978 q^{45} +2.45322 q^{46} -3.56238 q^{47} -4.01431 q^{48} -5.44502 q^{49} +4.40401 q^{50} -0.804154 q^{51} +1.48713 q^{52} +4.43853 q^{53} +7.34407 q^{54} +1.55515 q^{55} +2.31804 q^{56} +1.00556 q^{57} -9.79807 q^{58} +6.48066 q^{59} -1.13847 q^{60} -8.45447 q^{61} -4.51288 q^{62} -2.93459 q^{63} +1.79802 q^{64} -2.54044 q^{65} +1.37187 q^{66} +1.04951 q^{67} -0.910358 q^{68} -1.15639 q^{69} +3.30832 q^{70} +9.40662 q^{71} -4.37463 q^{72} -11.6804 q^{73} -10.4064 q^{74} -2.07594 q^{75} +1.13837 q^{76} -1.24699 q^{77} -2.24104 q^{78} -15.2522 q^{79} +7.76324 q^{80} +3.59820 q^{81} +12.8770 q^{82} +4.52363 q^{83} +0.912881 q^{84} +1.55515 q^{85} -1.70598 q^{86} +4.61856 q^{87} -1.85890 q^{88} +10.6280 q^{89} -6.24351 q^{90} +2.03705 q^{91} -1.30911 q^{92} +2.12725 q^{93} +6.07734 q^{94} -1.94465 q^{95} +3.85863 q^{96} +1.65291 q^{97} +9.28907 q^{98} +2.35334 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 66 q + 7 q^{2} + 3 q^{3} + 61 q^{4} + 4 q^{5} + 10 q^{6} + 14 q^{7} + 21 q^{8} + 65 q^{9}+O(q^{10}) $ 66 * q + 7 * q^2 + 3 * q^3 + 61 * q^4 + 4 * q^5 + 10 * q^6 + 14 * q^7 + 21 * q^8 + 65 * q^9	\( 66 q + 7 q^{2} + 3 q^{3} + 61 q^{4} + 4 q^{5} + 10 q^{6} + 14 q^{7} + 21 q^{8} + 65 q^{9} + 13 q^{10} - 66 q^{11} + 9 q^{12} - 12 q^{13} + 25 q^{14} + 13 q^{15} + 47 q^{16} - 66 q^{17} + 37 q^{18} + 19 q^{20} + 26 q^{21} - 7 q^{22} + 47 q^{23} + 15 q^{24} + 52 q^{25} + 16 q^{26} + 9 q^{27} + 3 q^{28} + 57 q^{29} + 2 q^{30} + 31 q^{31} + 39 q^{32} - 3 q^{33} - 7 q^{34} + 36 q^{35} + 39 q^{36} - 14 q^{37} + 18 q^{38} + 71 q^{39} + 29 q^{40} + 62 q^{41} - 3 q^{42} + 66 q^{43} - 61 q^{44} - 2 q^{45} + 19 q^{46} + 32 q^{47} + 26 q^{48} + 42 q^{49} + 10 q^{50} - 3 q^{51} - 7 q^{52} + 33 q^{53} + 100 q^{54} - 4 q^{55} + 61 q^{56} + 35 q^{57} - 16 q^{58} + 59 q^{59} + 50 q^{60} + 26 q^{61} + 29 q^{62} + 62 q^{63} + 29 q^{64} + 55 q^{65} - 10 q^{66} + 5 q^{67} - 61 q^{68} - 36 q^{69} - 35 q^{70} + 128 q^{71} + 87 q^{72} + 23 q^{73} + 64 q^{74} - 11 q^{75} + 74 q^{76} - 14 q^{77} + 45 q^{78} + 39 q^{79} + 95 q^{80} + 54 q^{81} - 6 q^{82} + 48 q^{83} + 38 q^{84} - 4 q^{85} + 7 q^{86} + 14 q^{87} - 21 q^{88} + 28 q^{89} + 135 q^{90} - 18 q^{91} + 108 q^{92} - 9 q^{93} + 37 q^{94} + 149 q^{95} + 104 q^{96} + 19 q^{97} + 30 q^{98} - 65 q^{99}+O(q^{100}) \) 66 * q + 7 * q^2 + 3 * q^3 + 61 * q^4 + 4 * q^5 + 10 * q^6 + 14 * q^7 + 21 * q^8 + 65 * q^9 + 13 * q^10 - 66 * q^11 + 9 * q^12 - 12 * q^13 + 25 * q^14 + 13 * q^15 + 47 * q^16 - 66 * q^17 + 37 * q^18 + 19 * q^20 + 26 * q^21 - 7 * q^22 + 47 * q^23 + 15 * q^24 + 52 * q^25 + 16 * q^26 + 9 * q^27 + 3 * q^28 + 57 * q^29 + 2 * q^30 + 31 * q^31 + 39 * q^32 - 3 * q^33 - 7 * q^34 + 36 * q^35 + 39 * q^36 - 14 * q^37 + 18 * q^38 + 71 * q^39 + 29 * q^40 + 62 * q^41 - 3 * q^42 + 66 * q^43 - 61 * q^44 - 2 * q^45 + 19 * q^46 + 32 * q^47 + 26 * q^48 + 42 * q^49 + 10 * q^50 - 3 * q^51 - 7 * q^52 + 33 * q^53 + 100 * q^54 - 4 * q^55 + 61 * q^56 + 35 * q^57 - 16 * q^58 + 59 * q^59 + 50 * q^60 + 26 * q^61 + 29 * q^62 + 62 * q^63 + 29 * q^64 + 55 * q^65 - 10 * q^66 + 5 * q^67 - 61 * q^68 - 36 * q^69 - 35 * q^70 + 128 * q^71 + 87 * q^72 + 23 * q^73 + 64 * q^74 - 11 * q^75 + 74 * q^76 - 14 * q^77 + 45 * q^78 + 39 * q^79 + 95 * q^80 + 54 * q^81 - 6 * q^82 + 48 * q^83 + 38 * q^84 - 4 * q^85 + 7 * q^86 + 14 * q^87 - 21 * q^88 + 28 * q^89 + 135 * q^90 - 18 * q^91 + 108 * q^92 - 9 * q^93 + 37 * q^94 + 149 * q^95 + 104 * q^96 + 19 * q^97 + 30 * q^98 - 65 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.70598	−1.20631	−0.603154	−	0.797625i	$-0.706090\pi$
$2$	−1.70598	−1.20631	−0.603154	+	0.797625i	$0.706090\pi$
$3$	0.804154	0.464278	0.232139	−	0.972683i	$-0.425427\pi$
$3$	0.804154	0.464278	0.232139	+	0.972683i	$0.425427\pi$
$4$	0.910358	0.455179
$5$	−1.55515	−0.695483	−0.347741	−	0.937590i	$-0.613051\pi$
$5$	−1.55515	−0.695483	−0.347741	+	0.937590i	$0.613051\pi$
$6$	−1.37187	−0.560063
$7$	1.24699	0.471318	0.235659	−	0.971836i	$-0.424275\pi$
$7$	1.24699	0.471318	0.235659	+	0.971836i	$0.424275\pi$
$8$	1.85890	0.657222
$9$	−2.35334	−0.784445
$10$	2.65305	0.838967
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.732068	0.211330
$13$	1.63357	0.453071	0.226535	−	0.974003i	$-0.427260\pi$
$13$	1.63357	0.453071	0.226535	+	0.974003i	$0.427260\pi$
$14$	−2.12734	−0.568555
$15$	−1.25058	−0.322898
$16$	−4.99196	−1.24799
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	4.01474	0.946283
$19$	1.25046	0.286875	0.143438	−	0.989659i	$-0.454184\pi$
$19$	1.25046	0.286875	0.143438	+	0.989659i	$0.454184\pi$
$20$	−1.41574	−0.316569
$21$	1.00277	0.218823
$22$	1.70598	0.363716
$23$	−1.43802	−0.299847	−0.149923	−	0.988698i	$-0.547903\pi$
$23$	−1.43802	−0.299847	−0.149923	+	0.988698i	$0.547903\pi$
$24$	1.49485	0.305134
$25$	−2.58152	−0.516304
$26$	−2.78683	−0.546543
$27$	−4.30491	−0.828480
$28$	1.13521	0.214534
$29$	5.74338	1.06652	0.533259	−	0.845952i	$-0.320967\pi$
$29$	5.74338	1.06652	0.533259	+	0.845952i	$0.320967\pi$
$30$	2.13346	0.389514
$31$	2.64533	0.475116	0.237558	−	0.971373i	$-0.423653\pi$
$31$	2.64533	0.475116	0.237558	+	0.971373i	$0.423653\pi$
$32$	4.79837	0.848240
$33$	−0.804154	−0.139985
$34$	1.70598	0.292573
$35$	−1.93925	−0.327794
$36$	−2.14238	−0.357063
$37$	6.09996	1.00283	0.501414	−	0.865208i	$-0.332813\pi$
$37$	6.09996	1.00283	0.501414	+	0.865208i	$0.332813\pi$
$38$	−2.13326	−0.346060
$39$	1.31364	0.210351
$40$	−2.89087	−0.457087
$41$	−7.54817	−1.17883	−0.589413	−	0.807832i	$-0.700641\pi$
$41$	−7.54817	−1.17883	−0.589413	+	0.807832i	$0.700641\pi$
$42$	−1.71071	−0.263968
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	−0.910358	−0.137242
$45$	3.65978	0.545568
$46$	2.45322	0.361708
$47$	−3.56238	−0.519626	−0.259813	−	0.965659i	$-0.583661\pi$
$47$	−3.56238	−0.519626	−0.259813	+	0.965659i	$0.583661\pi$
$48$	−4.01431	−0.579415
$49$	−5.44502	−0.777859
$50$	4.40401	0.622821
$51$	−0.804154	−0.112604
$52$	1.48713	0.206228
$53$	4.43853	0.609679	0.304839	−	0.952404i	$-0.401397\pi$
$53$	4.43853	0.609679	0.304839	+	0.952404i	$0.401397\pi$
$54$	7.34407	0.999402
$55$	1.55515	0.209696
$56$	2.31804	0.309760
$57$	1.00556	0.133190
$58$	−9.79807	−1.28655
$59$	6.48066	0.843710	0.421855	−	0.906663i	$-0.361379\pi$
$59$	6.48066	0.843710	0.421855	+	0.906663i	$0.361379\pi$
$60$	−1.13847	−0.146976
$61$	−8.45447	−1.08248	−0.541242	−	0.840867i	$-0.682046\pi$
$61$	−8.45447	−1.08248	−0.541242	+	0.840867i	$0.682046\pi$
$62$	−4.51288	−0.573136
$63$	−2.93459	−0.369723
$64$	1.79802	0.224753
$65$	−2.54044	−0.315103
$66$	1.37187	0.168865
$67$	1.04951	0.128218	0.0641092	−	0.997943i	$-0.479579\pi$
$67$	1.04951	0.128218	0.0641092	+	0.997943i	$0.479579\pi$
$68$	−0.910358	−0.110397
$69$	−1.15639	−0.139212
$70$	3.30832	0.395420
$71$	9.40662	1.11636	0.558180	−	0.829720i	$-0.311500\pi$
$71$	9.40662	1.11636	0.558180	+	0.829720i	$0.311500\pi$
$72$	−4.37463	−0.515555
$73$	−11.6804	−1.36709	−0.683546	−	0.729908i	$-0.739563\pi$
$73$	−11.6804	−1.36709	−0.683546	+	0.729908i	$0.739563\pi$
$74$	−10.4064	−1.20972
$75$	−2.07594	−0.239709
$76$	1.13837	0.130580
$77$	−1.24699	−0.142108
$78$	−2.24104	−0.253748
$79$	−15.2522	−1.71601	−0.858005	−	0.513642i	$-0.828296\pi$
$79$	−15.2522	−1.71601	−0.858005	+	0.513642i	$0.828296\pi$
$80$	7.76324	0.867956
$81$	3.59820	0.399800
$82$	12.8770	1.42203
$83$	4.52363	0.496533	0.248267	−	0.968692i	$-0.420139\pi$
$83$	4.52363	0.496533	0.248267	+	0.968692i	$0.420139\pi$
$84$	0.912881	0.0996035
$85$	1.55515	0.168679
$86$	−1.70598	−0.183960
$87$	4.61856	0.495161
$88$	−1.85890	−0.198160
$89$	10.6280	1.12657	0.563283	−	0.826264i	$-0.309538\pi$
$89$	10.6280	1.12657	0.563283	+	0.826264i	$0.309538\pi$
$90$	−6.24351	−0.658124
$91$	2.03705	0.213540
$92$	−1.30911	−0.136484
$93$	2.12725	0.220586
$94$	6.07734	0.626830
$95$	−1.94465	−0.199517
$96$	3.85863	0.393819
$97$	1.65291	0.167828	0.0839138	−	0.996473i	$-0.473258\pi$
$97$	1.65291	0.167828	0.0839138	+	0.996473i	$0.473258\pi$
$98$	9.28907	0.938338
$99$	2.35334	0.236519
$100$	−2.35010	−0.235010
$101$	−8.46000	−0.841802	−0.420901	−	0.907107i	$-0.638286\pi$
$101$	−8.46000	−0.841802	−0.420901	+	0.907107i	$0.638286\pi$
$102$	1.37187	0.135835
$103$	−11.6456	−1.14748	−0.573740	−	0.819038i	$-0.694508\pi$
$103$	−11.6456	−1.14748	−0.573740	+	0.819038i	$0.694508\pi$
$104$	3.03665	0.297768
$105$	−1.55946	−0.152187
$106$	−7.57203	−0.735461
$107$	6.43639	0.622229	0.311115	−	0.950372i	$-0.399298\pi$
$107$	6.43639	0.622229	0.311115	+	0.950372i	$0.399298\pi$
$108$	−3.91901	−0.377106
$109$	10.8526	1.03949	0.519746	−	0.854321i	$-0.326026\pi$
$109$	10.8526	1.03949	0.519746	+	0.854321i	$0.326026\pi$
$110$	−2.65305	−0.252958
$111$	4.90531	0.465591
$112$	−6.22493	−0.588201
$113$	6.59244	0.620165	0.310082	−	0.950710i	$-0.399643\pi$
$113$	6.59244	0.620165	0.310082	+	0.950710i	$0.399643\pi$
$114$	−1.71547	−0.160668
$115$	2.23633	0.208538
$116$	5.22853	0.485457
$117$	−3.84434	−0.355409
$118$	−11.0559	−1.01777
$119$	−1.24699	−0.114311
$120$	−2.32470	−0.212215
$121$	1.00000	0.0909091
$122$	14.4231	1.30581
$123$	−6.06989	−0.547304
$124$	2.40820	0.216263
$125$	11.7904	1.05456
$126$	5.00634	0.446000
$127$	12.1849	1.08124	0.540619	−	0.841268i	$-0.318190\pi$
$127$	12.1849	1.08124	0.540619	+	0.841268i	$0.318190\pi$
$128$	−12.6641	−1.11936
$129$	0.804154	0.0708018
$130$	4.33393	0.380111
$131$	−4.29951	−0.375650	−0.187825	−	0.982203i	$-0.560144\pi$
$131$	−4.29951	−0.375650	−0.187825	+	0.982203i	$0.560144\pi$
$132$	−0.732068	−0.0637183
$133$	1.55931	0.135210
$134$	−1.79045	−0.154671
$135$	6.69476	0.576193
$136$	−1.85890	−0.159400
$137$	1.60905	0.137470	0.0687351	−	0.997635i	$-0.478104\pi$
$137$	1.60905	0.137470	0.0687351	+	0.997635i	$0.478104\pi$
$138$	1.97277	0.167933
$139$	−3.31148	−0.280876	−0.140438	−	0.990089i	$-0.544851\pi$
$139$	−3.31148	−0.280876	−0.140438	+	0.990089i	$0.544851\pi$
$140$	−1.76541	−0.149205
$141$	−2.86470	−0.241251
$142$	−16.0475	−1.34667
$143$	−1.63357	−0.136606
$144$	11.7478	0.978981
$145$	−8.93179	−0.741745
$146$	19.9266	1.64913
$147$	−4.37863	−0.361143
$148$	5.55315	0.456466
$149$	−1.07492	−0.0880605	−0.0440302	−	0.999030i	$-0.514020\pi$
$149$	−1.07492	−0.0880605	−0.0440302	+	0.999030i	$0.514020\pi$
$150$	3.54150	0.289162
$151$	−1.39912	−0.113859	−0.0569296	−	0.998378i	$-0.518131\pi$
$151$	−1.39912	−0.113859	−0.0569296	+	0.998378i	$0.518131\pi$
$152$	2.32449	0.188541
$153$	2.35334	0.190256
$154$	2.12734	0.171426
$155$	−4.11388	−0.330435
$156$	1.19588	0.0957473
$157$	−5.67744	−0.453109	−0.226555	−	0.973998i	$-0.572746\pi$
$157$	−5.67744	−0.453109	−0.226555	+	0.973998i	$0.572746\pi$
$158$	26.0199	2.07004
$159$	3.56926	0.283061
$160$	−7.46217	−0.589936
$161$	−1.79319	−0.141323
$162$	−6.13845	−0.482282
$163$	−9.50915	−0.744814	−0.372407	−	0.928069i	$-0.621468\pi$
$163$	−9.50915	−0.744814	−0.372407	+	0.928069i	$0.621468\pi$
$164$	−6.87154	−0.536577
$165$	1.25058	0.0973573
$166$	−7.71722	−0.598972
$167$	−16.4072	−1.26963	−0.634815	−	0.772664i	$-0.718924\pi$
$167$	−16.4072	−1.26963	−0.634815	+	0.772664i	$0.718924\pi$
$168$	1.86406	0.143815
$169$	−10.3314	−0.794727
$170$	−2.65305	−0.203479
$171$	−2.94276	−0.225038
$172$	0.910358	0.0694141
$173$	22.3235	1.69722	0.848610	−	0.529018i	$-0.177440\pi$
$173$	22.3235	1.69722	0.848610	+	0.529018i	$0.177440\pi$
$174$	−7.87915	−0.597317
$175$	−3.21913	−0.243343
$176$	4.99196	0.376283
$177$	5.21145	0.391716
$178$	−18.1311	−1.35898
$179$	19.9179	1.48873	0.744367	−	0.667771i	$-0.232751\pi$
$179$	19.9179	1.48873	0.744367	+	0.667771i	$0.232751\pi$
$180$	3.33171	0.248331
$181$	18.6994	1.38992	0.694958	−	0.719050i	$-0.255423\pi$
$181$	18.6994	1.38992	0.694958	+	0.719050i	$0.255423\pi$
$182$	−3.47515	−0.257595
$183$	−6.79870	−0.502574
$184$	−2.67313	−0.197066
$185$	−9.48634	−0.697449
$186$	−3.62905	−0.266095
$187$	1.00000	0.0731272
$188$	−3.24304	−0.236523
$189$	−5.36817	−0.390477
$190$	3.31753	0.240679
$191$	−3.85520	−0.278953	−0.139476	−	0.990225i	$-0.544542\pi$
$191$	−3.85520	−0.278953	−0.139476	+	0.990225i	$0.544542\pi$
$192$	1.44589	0.104348
$193$	−2.57308	−0.185214	−0.0926070	−	0.995703i	$-0.529520\pi$
$193$	−2.57308	−0.185214	−0.0926070	+	0.995703i	$0.529520\pi$
$194$	−2.81983	−0.202452
$195$	−2.04291	−0.146296
$196$	−4.95691	−0.354065
$197$	−4.47229	−0.318637	−0.159319	−	0.987227i	$-0.550930\pi$
$197$	−4.47229	−0.318637	−0.159319	+	0.987227i	$0.550930\pi$
$198$	−4.01474	−0.285315
$199$	9.88305	0.700591	0.350296	−	0.936639i	$-0.386081\pi$
$199$	9.88305	0.700591	0.350296	+	0.936639i	$0.386081\pi$
$200$	−4.79879	−0.339326
$201$	0.843970	0.0595291
$202$	14.4326	1.01547
$203$	7.16193	0.502669
$204$	−0.732068	−0.0512550
$205$	11.7385	0.819854
$206$	19.8672	1.38421
$207$	3.38413	0.235214
$208$	−8.15472	−0.565428
$209$	−1.25046	−0.0864962
$210$	2.66040	0.183585
$211$	11.7923	0.811814	0.405907	−	0.913914i	$-0.366956\pi$
$211$	11.7923	0.811814	0.405907	+	0.913914i	$0.366956\pi$
$212$	4.04065	0.277513
$213$	7.56437	0.518302
$214$	−10.9803	−0.750600
$215$	−1.55515	−0.106060
$216$	−8.00241	−0.544495
$217$	3.29870	0.223931
$218$	−18.5143	−1.25395
$219$	−9.39287	−0.634711
$220$	1.41574	0.0954492
$221$	−1.63357	−0.109886
$222$	−8.36834	−0.561646
$223$	−16.8856	−1.13074	−0.565372	−	0.824836i	$-0.691267\pi$
$223$	−16.8856	−1.13074	−0.565372	+	0.824836i	$0.691267\pi$
$224$	5.98352	0.399790
$225$	6.07518	0.405012
$226$	−11.2466	−0.748109
$227$	24.8899	1.65200	0.826001	−	0.563669i	$-0.190611\pi$
$227$	24.8899	1.65200	0.826001	+	0.563669i	$0.190611\pi$
$228$	0.915422	0.0606253
$229$	−22.4009	−1.48029	−0.740147	−	0.672446i	$-0.765244\pi$
$229$	−22.4009	−1.48029	−0.740147	+	0.672446i	$0.765244\pi$
$230$	−3.81512	−0.251562
$231$	−1.00277	−0.0659775
$232$	10.6764	0.700939
$233$	4.68486	0.306915	0.153458	−	0.988155i	$-0.450959\pi$
$233$	4.68486	0.306915	0.153458	+	0.988155i	$0.450959\pi$
$234$	6.55836	0.428733
$235$	5.54002	0.361391
$236$	5.89972	0.384039
$237$	−12.2651	−0.796706
$238$	2.12734	0.137895
$239$	−29.8551	−1.93117	−0.965583	−	0.260093i	$-0.916247\pi$
$239$	−29.8551	−1.93117	−0.965583	+	0.260093i	$0.916247\pi$
$240$	6.24284	0.402974
$241$	−14.3137	−0.922028	−0.461014	−	0.887393i	$-0.652514\pi$
$241$	−14.3137	−0.922028	−0.461014	+	0.887393i	$0.652514\pi$
$242$	−1.70598	−0.109664
$243$	15.8082	1.01410
$244$	−7.69660	−0.492724
$245$	8.46780	0.540988
$246$	10.3551	0.660217
$247$	2.04272	0.129975
$248$	4.91742	0.312257
$249$	3.63770	0.230530
$250$	−20.1141	−1.27213
$251$	−3.72735	−0.235268	−0.117634	−	0.993057i	$-0.537531\pi$
$251$	−3.72735	−0.235268	−0.117634	+	0.993057i	$0.537531\pi$
$252$	−2.67152	−0.168290
$253$	1.43802	0.0904073
$254$	−20.7872	−1.30431
$255$	1.25058	0.0783142
$256$	18.0087	1.12554
$257$	12.1038	0.755017	0.377508	−	0.926006i	$-0.376781\pi$
$257$	12.1038	0.755017	0.377508	+	0.926006i	$0.376781\pi$
$258$	−1.37187	−0.0854088
$259$	7.60659	0.472651
$260$	−2.31271	−0.143428
$261$	−13.5161	−0.836625
$262$	7.33487	0.453150
$263$	28.5809	1.76237	0.881186	−	0.472769i	$-0.156745\pi$
$263$	28.5809	1.76237	0.881186	+	0.472769i	$0.156745\pi$
$264$	−1.49485	−0.0920014
$265$	−6.90257	−0.424021
$266$	−2.66015	−0.163104
$267$	8.54654	0.523040
$268$	0.955432	0.0583623
$269$	−22.0896	−1.34683	−0.673414	−	0.739266i	$-0.735173\pi$
$269$	−22.0896	−1.34683	−0.673414	+	0.739266i	$0.735173\pi$
$270$	−11.4211	−0.695067
$271$	2.34716	0.142580	0.0712898	−	0.997456i	$-0.477288\pi$
$271$	2.34716	0.142580	0.0712898	+	0.997456i	$0.477288\pi$
$272$	4.99196	0.302682
$273$	1.63810	0.0991422
$274$	−2.74500	−0.165831
$275$	2.58152	0.155671
$276$	−1.05272	−0.0633666
$277$	30.6531	1.84177	0.920884	−	0.389836i	$-0.127468\pi$
$277$	30.6531	1.84177	0.920884	+	0.389836i	$0.127468\pi$
$278$	5.64930	0.338823
$279$	−6.22536	−0.372702
$280$	−3.60489	−0.215433
$281$	17.0121	1.01486	0.507429	−	0.861693i	$-0.330596\pi$
$281$	17.0121	1.01486	0.507429	+	0.861693i	$0.330596\pi$
$282$	4.88711	0.291023
$283$	−17.0181	−1.01162	−0.505810	−	0.862645i	$-0.668806\pi$
$283$	−17.0181	−1.01162	−0.505810	+	0.862645i	$0.668806\pi$
$284$	8.56339	0.508144
$285$	−1.56380	−0.0926314
$286$	2.78683	0.164789
$287$	−9.41250	−0.555602
$288$	−11.2922	−0.665398
$289$	1.00000	0.0588235
$290$	15.2374	0.894773
$291$	1.32919	0.0779187
$292$	−10.6334	−0.622271
$293$	−9.60369	−0.561053	−0.280527	−	0.959846i	$-0.590509\pi$
$293$	−9.60369	−0.561053	−0.280527	+	0.959846i	$0.590509\pi$
$294$	7.46984	0.435650
$295$	−10.0784	−0.586786
$296$	11.3392	0.659080
$297$	4.30491	0.249796
$298$	1.83378	0.106228
$299$	−2.34910	−0.135852
$300$	−1.88985	−0.109110
$301$	1.24699	0.0718753
$302$	2.38688	0.137349
$303$	−6.80314	−0.390830
$304$	−6.24226	−0.358018
$305$	13.1479	0.752849
$306$	−4.01474	−0.229507
$307$	34.0200	1.94162	0.970812	−	0.239844i	$-0.0770961\pi$
$307$	34.0200	1.94162	0.970812	+	0.239844i	$0.0770961\pi$
$308$	−1.13521	−0.0646844
$309$	−9.36489	−0.532750
$310$	7.01819	0.398606
$311$	−32.2558	−1.82906	−0.914529	−	0.404521i	$-0.867438\pi$
$311$	−32.2558	−1.82906	−0.914529	+	0.404521i	$0.867438\pi$
$312$	2.44193	0.138247
$313$	27.8922	1.57656	0.788281	−	0.615316i	$-0.210972\pi$
$313$	27.8922	1.57656	0.788281	+	0.615316i	$0.210972\pi$
$314$	9.68559	0.546590
$315$	4.56371	0.257136
$316$	−13.8850	−0.781091
$317$	−18.1463	−1.01920	−0.509600	−	0.860411i	$-0.670207\pi$
$317$	−18.1463	−1.01920	−0.509600	+	0.860411i	$0.670207\pi$
$318$	−6.08908	−0.341459
$319$	−5.74338	−0.321567
$320$	−2.79619	−0.156312
$321$	5.17585	0.288888
$322$	3.05914	0.170479
$323$	−1.25046	−0.0695775
$324$	3.27565	0.181981
$325$	−4.21709	−0.233922
$326$	16.2224	0.898476
$327$	8.72718	0.482614
$328$	−14.0313	−0.774751
$329$	−4.44225	−0.244909
$330$	−2.13346	−0.117443
$331$	−2.65857	−0.146128	−0.0730642	−	0.997327i	$-0.523278\pi$
$331$	−2.65857	−0.146128	−0.0730642	+	0.997327i	$0.523278\pi$
$332$	4.11813	0.226011
$333$	−14.3553	−0.786664
$334$	27.9904	1.53157
$335$	−1.63215	−0.0891737
$336$	−5.00580	−0.273089
$337$	2.31326	0.126011	0.0630057	−	0.998013i	$-0.479931\pi$
$337$	2.31326	0.126011	0.0630057	+	0.998013i	$0.479931\pi$
$338$	17.6252	0.958685
$339$	5.30134	0.287929
$340$	1.41574	0.0767793
$341$	−2.64533	−0.143253
$342$	5.02027	0.271465
$343$	−15.5188	−0.837937
$344$	1.85890	0.100225
$345$	1.79835	0.0968199
$346$	−38.0833	−2.04737
$347$	21.8938	1.17532	0.587660	−	0.809108i	$-0.300049\pi$
$347$	21.8938	1.17532	0.587660	+	0.809108i	$0.300049\pi$
$348$	4.20454	0.225387
$349$	−4.93814	−0.264333	−0.132166	−	0.991228i	$-0.542193\pi$
$349$	−4.93814	−0.264333	−0.132166	+	0.991228i	$0.542193\pi$
$350$	5.49176	0.293547
$351$	−7.03237	−0.375360
$352$	−4.79837	−0.255754
$353$	−28.1063	−1.49595	−0.747973	−	0.663729i	$-0.768973\pi$
$353$	−28.1063	−1.49595	−0.747973	+	0.663729i	$0.768973\pi$
$354$	−8.89061	−0.472531
$355$	−14.6287	−0.776410
$356$	9.67528	0.512789
$357$	−1.00277	−0.0530723
$358$	−33.9795	−1.79587
$359$	−4.56365	−0.240860	−0.120430	−	0.992722i	$-0.538427\pi$
$359$	−4.56365	−0.240860	−0.120430	+	0.992722i	$0.538427\pi$
$360$	6.80319	0.358560
$361$	−17.4363	−0.917702
$362$	−31.9008	−1.67667
$363$	0.804154	0.0422071
$364$	1.85444	0.0971991
$365$	18.1648	0.950789
$366$	11.5984	0.606259
$367$	16.4416	0.858245	0.429123	−	0.903246i	$-0.358823\pi$
$367$	16.4416	0.858245	0.429123	+	0.903246i	$0.358823\pi$
$368$	7.17852	0.374206
$369$	17.7634	0.924725
$370$	16.1835	0.841339
$371$	5.53480	0.287353
$372$	1.93656	0.100406
$373$	5.14329	0.266309	0.133155	−	0.991095i	$-0.457489\pi$
$373$	5.14329	0.266309	0.133155	+	0.991095i	$0.457489\pi$
$374$	−1.70598	−0.0882140
$375$	9.48128	0.489611
$376$	−6.62212	−0.341510
$377$	9.38221	0.483208
$378$	9.15798	0.471036
$379$	−14.4574	−0.742625	−0.371312	−	0.928508i	$-0.621092\pi$
$379$	−14.4574	−0.742625	−0.371312	+	0.928508i	$0.621092\pi$
$380$	−1.77033	−0.0908159
$381$	9.79856	0.501996
$382$	6.57689	0.336503
$383$	33.1972	1.69630	0.848148	−	0.529760i	$-0.177718\pi$
$383$	33.1972	1.69630	0.848148	+	0.529760i	$0.177718\pi$
$384$	−10.1839	−0.519695
$385$	1.93925	0.0988335
$386$	4.38961	0.223425
$387$	−2.35334	−0.119627
$388$	1.50474	0.0763916
$389$	19.6996	0.998811	0.499406	−	0.866368i	$-0.333552\pi$
$389$	19.6996	0.998811	0.499406	+	0.866368i	$0.333552\pi$
$390$	3.48515	0.176477
$391$	1.43802	0.0727236
$392$	−10.1218	−0.511226
$393$	−3.45747	−0.174406
$394$	7.62962	0.384375
$395$	23.7194	1.19346
$396$	2.14238	0.107659
$397$	4.15674	0.208621	0.104310	−	0.994545i	$-0.466736\pi$
$397$	4.15674	0.208621	0.104310	+	0.994545i	$0.466736\pi$
$398$	−16.8603	−0.845129
$399$	1.25393	0.0627749
$400$	12.8868	0.644342
$401$	5.80193	0.289735	0.144867	−	0.989451i	$-0.453724\pi$
$401$	5.80193	0.289735	0.144867	+	0.989451i	$0.453724\pi$
$402$	−1.43979	−0.0718104
$403$	4.32134	0.215261
$404$	−7.70163	−0.383170
$405$	−5.59573	−0.278054
$406$	−12.2181	−0.606374
$407$	−6.09996	−0.302364
$408$	−1.49485	−0.0740059
$409$	32.2101	1.59269	0.796345	−	0.604843i	$-0.206764\pi$
$409$	32.2101	1.59269	0.796345	+	0.604843i	$0.206764\pi$
$410$	−20.0256	−0.988996
$411$	1.29392	0.0638244
$412$	−10.6017	−0.522309
$413$	8.08132	0.397656
$414$	−5.77326	−0.283740
$415$	−7.03492	−0.345330
$416$	7.83847	0.384313
$417$	−2.66294	−0.130405
$418$	2.13326	0.104341
$419$	30.4883	1.48945	0.744727	−	0.667370i	$-0.232580\pi$
$419$	30.4883	1.48945	0.744727	+	0.667370i	$0.232580\pi$
$420$	−1.41966	−0.0692725
$421$	21.8335	1.06410	0.532050	−	0.846713i	$-0.321422\pi$
$421$	21.8335	1.06410	0.532050	+	0.846713i	$0.321422\pi$
$422$	−20.1173	−0.979297
$423$	8.38348	0.407619
$424$	8.25080	0.400694
$425$	2.58152	0.125222
$426$	−12.9046	−0.625232
$427$	−10.5426	−0.510194
$428$	5.85942	0.283226
$429$	−1.31364	−0.0634232
$430$	2.65305	0.127941
$431$	8.83929	0.425774	0.212887	−	0.977077i	$-0.431713\pi$
$431$	8.83929	0.425774	0.212887	+	0.977077i	$0.431713\pi$
$432$	21.4899	1.03394
$433$	22.3906	1.07602	0.538012	−	0.842937i	$-0.319176\pi$
$433$	22.3906	1.07602	0.538012	+	0.842937i	$0.319176\pi$
$434$	−5.62751	−0.270129
$435$	−7.18254	−0.344376
$436$	9.87977	0.473155
$437$	−1.79818	−0.0860187
$438$	16.0240	0.765657
$439$	−8.78513	−0.419292	−0.209646	−	0.977777i	$-0.567231\pi$
$439$	−8.78513	−0.419292	−0.209646	+	0.977777i	$0.567231\pi$
$440$	2.89087	0.137817
$441$	12.8140	0.610188
$442$	2.78683	0.132556
$443$	8.80962	0.418558	0.209279	−	0.977856i	$-0.432888\pi$
$443$	8.80962	0.418558	0.209279	+	0.977856i	$0.432888\pi$
$444$	4.46559	0.211927
$445$	−16.5281	−0.783507
$446$	28.8065	1.36403
$447$	−0.864397	−0.0408846
$448$	2.24212	0.105930
$449$	13.7657	0.649645	0.324823	−	0.945775i	$-0.394695\pi$
$449$	13.7657	0.649645	0.324823	+	0.945775i	$0.394695\pi$
$450$	−10.3641	−0.488569
$451$	7.54817	0.355430
$452$	6.00148	0.282286
$453$	−1.12511	−0.0528624
$454$	−42.4616	−1.99282
$455$	−3.16790	−0.148514
$456$	1.86925	0.0875355
$457$	−17.6050	−0.823529	−0.411765	−	0.911290i	$-0.635087\pi$
$457$	−17.6050	−0.823529	−0.411765	+	0.911290i	$0.635087\pi$
$458$	38.2154	1.78569
$459$	4.30491	0.200936
$460$	2.03586	0.0949223
$461$	40.2746	1.87577	0.937887	−	0.346941i	$-0.112779\pi$
$461$	40.2746	1.87577	0.937887	+	0.346941i	$0.112779\pi$
$462$	1.71071	0.0795892
$463$	27.5791	1.28171	0.640854	−	0.767663i	$-0.278580\pi$
$463$	27.5791	1.28171	0.640854	+	0.767663i	$0.278580\pi$
$464$	−28.6707	−1.33101
$465$	−3.30819	−0.153414
$466$	−7.99226	−0.370234
$467$	2.82868	0.130896	0.0654479	−	0.997856i	$-0.479152\pi$
$467$	2.82868	0.130896	0.0654479	+	0.997856i	$0.479152\pi$
$468$	−3.49972	−0.161775
$469$	1.30873	0.0604316
$470$	−9.45115	−0.435949
$471$	−4.56554	−0.210369
$472$	12.0469	0.554505
$473$	−1.00000	−0.0459800
$474$	20.9240	0.961073
$475$	−3.22809	−0.148115
$476$	−1.13521	−0.0520321
$477$	−10.4454	−0.478260
$478$	50.9321	2.32958
$479$	−25.9089	−1.18381	−0.591903	−	0.806009i	$-0.701623\pi$
$479$	−25.9089	−1.18381	−0.591903	+	0.806009i	$0.701623\pi$
$480$	−6.00073	−0.273895
$481$	9.96471	0.454352
$482$	24.4189	1.11225
$483$	−1.44200	−0.0656133
$484$	0.910358	0.0413799
$485$	−2.57052	−0.116721
$486$	−26.9685	−1.22331
$487$	0.707061	0.0320400	0.0160200	−	0.999872i	$-0.494900\pi$
$487$	0.707061	0.0320400	0.0160200	+	0.999872i	$0.494900\pi$
$488$	−15.7161	−0.711432
$489$	−7.64682	−0.345801
$490$	−14.4459	−0.652598
$491$	16.8348	0.759742	0.379871	−	0.925039i	$-0.375968\pi$
$491$	16.8348	0.759742	0.379871	+	0.925039i	$0.375968\pi$
$492$	−5.52578	−0.249121
$493$	−5.74338	−0.258669
$494$	−3.48483	−0.156790
$495$	−3.65978	−0.164495
$496$	−13.2054	−0.592940
$497$	11.7300	0.526161
$498$	−6.20583	−0.278090
$499$	27.8890	1.24848	0.624242	−	0.781231i	$-0.285408\pi$
$499$	27.8890	1.24848	0.624242	+	0.781231i	$0.285408\pi$
$500$	10.7335	0.480015
$501$	−13.1939	−0.589462
$502$	6.35878	0.283806
$503$	19.4236	0.866055	0.433027	−	0.901381i	$-0.357445\pi$
$503$	19.4236	0.866055	0.433027	+	0.901381i	$0.357445\pi$
$504$	−5.45512	−0.242990
$505$	13.1565	0.585459
$506$	−2.45322	−0.109059
$507$	−8.30808	−0.368975
$508$	11.0927	0.492157
$509$	−35.0924	−1.55544	−0.777721	−	0.628609i	$-0.783624\pi$
$509$	−35.0924	−1.55544	−0.777721	+	0.628609i	$0.783624\pi$
$510$	−2.13346	−0.0944711
$511$	−14.5654	−0.644335
$512$	−5.39411	−0.238388
$513$	−5.38312	−0.237670
$514$	−20.6489	−0.910783
$515$	18.1107	0.798052
$516$	0.732068	0.0322275
$517$	3.56238	0.156673
$518$	−12.9767	−0.570162
$519$	17.9515	0.787983
$520$	−4.72244	−0.207093
$521$	−4.48209	−0.196364	−0.0981819	−	0.995168i	$-0.531303\pi$
$521$	−4.48209	−0.196364	−0.0981819	+	0.995168i	$0.531303\pi$
$522$	23.0582	1.00923
$523$	13.0670	0.571378	0.285689	−	0.958322i	$-0.407778\pi$
$523$	13.0670	0.571378	0.285689	+	0.958322i	$0.407778\pi$
$524$	−3.91409	−0.170988
$525$	−2.58867	−0.112979
$526$	−48.7583	−2.12596
$527$	−2.64533	−0.115233
$528$	4.01431	0.174700
$529$	−20.9321	−0.910092
$530$	11.7756	0.511500
$531$	−15.2512	−0.661845
$532$	1.41953	0.0615445
$533$	−12.3305	−0.534092
$534$	−14.5802	−0.630947
$535$	−10.0095	−0.432750
$536$	1.95094	0.0842680
$537$	16.0171	0.691187
$538$	37.6844	1.62469
$539$	5.44502	0.234533
$540$	6.09463	0.262271
$541$	−35.8083	−1.53952	−0.769760	−	0.638333i	$-0.779624\pi$
$541$	−35.8083	−1.53952	−0.769760	+	0.638333i	$0.779624\pi$
$542$	−4.00419	−0.171995
$543$	15.0372	0.645308
$544$	−4.79837	−0.205728
$545$	−16.8774	−0.722949
$546$	−2.79456	−0.119596
$547$	19.5598	0.836316	0.418158	−	0.908374i	$-0.362676\pi$
$547$	19.5598	0.836316	0.418158	+	0.908374i	$0.362676\pi$
$548$	1.46481	0.0625735
$549$	19.8962	0.849150
$550$	−4.40401	−0.187788
$551$	7.18187	0.305958
$552$	−2.14961	−0.0914935
$553$	−19.0194	−0.808786
$554$	−52.2936	−2.22174
$555$	−7.62848	−0.323811
$556$	−3.01463	−0.127849
$557$	32.1744	1.36327	0.681636	−	0.731692i	$-0.261269\pi$
$557$	32.1744	1.36327	0.681636	+	0.731692i	$0.261269\pi$
$558$	10.6203	0.449594
$559$	1.63357	0.0690926
$560$	9.68068	0.409083
$561$	0.804154	0.0339514
$562$	−29.0223	−1.22423
$563$	35.1304	1.48057	0.740284	−	0.672294i	$-0.234691\pi$
$563$	35.1304	1.48057	0.740284	+	0.672294i	$0.234691\pi$
$564$	−2.60790	−0.109813
$565$	−10.2522	−0.431314
$566$	29.0325	1.22033
$567$	4.48692	0.188433
$568$	17.4860	0.733697
$569$	0.809143	0.0339210	0.0169605	−	0.999856i	$-0.494601\pi$
$569$	0.809143	0.0339210	0.0169605	+	0.999856i	$0.494601\pi$
$570$	2.66780	0.111742
$571$	34.5294	1.44501	0.722505	−	0.691366i	$-0.242991\pi$
$571$	34.5294	1.44501	0.722505	+	0.691366i	$0.242991\pi$
$572$	−1.48713	−0.0621802
$573$	−3.10018	−0.129512
$574$	16.0575	0.670227
$575$	3.71226	0.154812
$576$	−4.23135	−0.176306
$577$	−0.741790	−0.0308811	−0.0154406	−	0.999881i	$-0.504915\pi$
$577$	−0.741790	−0.0308811	−0.0154406	+	0.999881i	$0.504915\pi$
$578$	−1.70598	−0.0709593
$579$	−2.06915	−0.0859909
$580$	−8.13113	−0.337627
$581$	5.64093	0.234025
$582$	−2.26757	−0.0939940
$583$	−4.43853	−0.183825
$584$	−21.7128	−0.898483
$585$	5.97851	0.247181
$586$	16.3837	0.676803
$587$	−14.7611	−0.609256	−0.304628	−	0.952471i	$-0.598532\pi$
$587$	−14.7611	−0.609256	−0.304628	+	0.952471i	$0.598532\pi$
$588$	−3.98612	−0.164385
$589$	3.30789	0.136299
$590$	17.1935	0.707845
$591$	−3.59641	−0.147936
$592$	−30.4508	−1.25152
$593$	−25.2259	−1.03591	−0.517953	−	0.855409i	$-0.673306\pi$
$593$	−25.2259	−1.03591	−0.517953	+	0.855409i	$0.673306\pi$
$594$	−7.34407	−0.301331
$595$	1.93925	0.0795016
$596$	−0.978558	−0.0400833
$597$	7.94750	0.325269
$598$	4.00751	0.163879
$599$	−17.2690	−0.705593	−0.352797	−	0.935700i	$-0.614769\pi$
$599$	−17.2690	−0.705593	−0.352797	+	0.935700i	$0.614769\pi$
$600$	−3.85897	−0.157542
$601$	−39.5852	−1.61471	−0.807356	−	0.590064i	$-0.799102\pi$
$601$	−39.5852	−1.61471	−0.807356	+	0.590064i	$0.799102\pi$
$602$	−2.12734	−0.0867038
$603$	−2.46986	−0.100580
$604$	−1.27370	−0.0518263
$605$	−1.55515	−0.0632257
$606$	11.6060	0.471462
$607$	−44.4651	−1.80478	−0.902391	−	0.430919i	$-0.858189\pi$
$607$	−44.4651	−1.80478	−0.902391	+	0.430919i	$0.858189\pi$
$608$	6.00017	0.243339
$609$	5.75930	0.233378
$610$	−22.4301	−0.908168
$611$	−5.81940	−0.235428
$612$	2.14238	0.0866005
$613$	8.00567	0.323346	0.161673	−	0.986844i	$-0.448311\pi$
$613$	8.00567	0.323346	0.161673	+	0.986844i	$0.448311\pi$
$614$	−58.0373	−2.34220
$615$	9.43958	0.380641
$616$	−2.31804	−0.0933963
$617$	48.0654	1.93504	0.967521	−	0.252793i	$-0.0813490\pi$
$617$	48.0654	1.93504	0.967521	+	0.252793i	$0.0813490\pi$
$618$	15.9763	0.642661
$619$	11.1238	0.447102	0.223551	−	0.974692i	$-0.428235\pi$
$619$	11.1238	0.447102	0.223551	+	0.974692i	$0.428235\pi$
$620$	−3.74510	−0.150407
$621$	6.19052	0.248417
$622$	55.0276	2.20641
$623$	13.2530	0.530970
$624$	−6.55765	−0.262516
$625$	−5.42818	−0.217127
$626$	−47.5835	−1.90182
$627$	−1.00556	−0.0401583
$628$	−5.16851	−0.206246
$629$	−6.09996	−0.243221
$630$	−7.78559	−0.310185
$631$	20.0317	0.797449	0.398725	−	0.917071i	$-0.369453\pi$
$631$	20.0317	0.797449	0.398725	+	0.917071i	$0.369453\pi$
$632$	−28.3524	−1.12780
$633$	9.48280	0.376908
$634$	30.9572	1.22947
$635$	−18.9494	−0.751983
$636$	3.24931	0.128843
$637$	−8.89481	−0.352425
$638$	9.79807	0.387909
$639$	−22.1369	−0.875724
$640$	19.6946	0.778496
$641$	41.4065	1.63546	0.817729	−	0.575604i	$-0.195233\pi$
$641$	41.4065	1.63546	0.817729	+	0.575604i	$0.195233\pi$
$642$	−8.82988	−0.348488
$643$	−29.1813	−1.15080	−0.575400	−	0.817872i	$-0.695153\pi$
$643$	−29.1813	−1.15080	−0.575400	+	0.817872i	$0.695153\pi$
$644$	−1.63245	−0.0643274
$645$	−1.25058	−0.0492414
$646$	2.13326	0.0839319
$647$	−24.6864	−0.970521	−0.485260	−	0.874370i	$-0.661275\pi$
$647$	−24.6864	−0.970521	−0.485260	+	0.874370i	$0.661275\pi$
$648$	6.68871	0.262758
$649$	−6.48066	−0.254388
$650$	7.19426	0.282182
$651$	2.65267	0.103966
$652$	−8.65673	−0.339024
$653$	36.5892	1.43185	0.715923	−	0.698179i	$-0.246006\pi$
$653$	36.5892	1.43185	0.715923	+	0.698179i	$0.246006\pi$
$654$	−14.8884	−0.582181
$655$	6.68637	0.261258
$656$	37.6802	1.47117
$657$	27.4880	1.07241
$658$	7.57838	0.295436
$659$	21.4657	0.836183	0.418092	−	0.908405i	$-0.362699\pi$
$659$	21.4657	0.836183	0.418092	+	0.908405i	$0.362699\pi$
$660$	1.13847	0.0443150
$661$	19.0812	0.742173	0.371087	−	0.928598i	$-0.378985\pi$
$661$	19.0812	0.742173	0.371087	+	0.928598i	$0.378985\pi$
$662$	4.53547	0.176276
$663$	−1.31364	−0.0510176
$664$	8.40900	0.326333
$665$	−2.42496	−0.0940359
$666$	24.4898	0.948959
$667$	−8.25906	−0.319792
$668$	−14.9365	−0.577909
$669$	−13.5786	−0.524980
$670$	2.78441	0.107571
$671$	8.45447	0.326381
$672$	4.81167	0.185614
$673$	−36.2090	−1.39575	−0.697877	−	0.716217i	$-0.745872\pi$
$673$	−36.2090	−1.39575	−0.697877	+	0.716217i	$0.745872\pi$
$674$	−3.94637	−0.152009
$675$	11.1132	0.427747
$676$	−9.40532	−0.361743
$677$	−2.12142	−0.0815328	−0.0407664	−	0.999169i	$-0.512980\pi$
$677$	−2.12142	−0.0815328	−0.0407664	+	0.999169i	$0.512980\pi$
$678$	−9.04396	−0.347331
$679$	2.06116	0.0791001
$680$	2.89087	0.110860
$681$	20.0153	0.766989
$682$	4.51288	0.172807
$683$	−19.2295	−0.735797	−0.367898	−	0.929866i	$-0.619923\pi$
$683$	−19.2295	−0.735797	−0.367898	+	0.929866i	$0.619923\pi$
$684$	−2.67896	−0.102433
$685$	−2.50231	−0.0956082
$686$	26.4747	1.01081
$687$	−18.0138	−0.687268
$688$	−4.99196	−0.190317
$689$	7.25065	0.276228
$690$	−3.06794	−0.116795
$691$	2.37270	0.0902619	0.0451309	−	0.998981i	$-0.485629\pi$
$691$	2.37270	0.0902619	0.0451309	+	0.998981i	$0.485629\pi$
$692$	20.3223	0.772539
$693$	2.93459	0.111476
$694$	−37.3503	−1.41780
$695$	5.14983	0.195344
$696$	8.58546	0.325431
$697$	7.54817	0.285908
$698$	8.42436	0.318867
$699$	3.76735	0.142494
$700$	−2.93056	−0.110765
$701$	48.0090	1.81328	0.906638	−	0.421909i	$-0.138640\pi$
$701$	48.0090	1.81328	0.906638	+	0.421909i	$0.138640\pi$
$702$	11.9971	0.452800
$703$	7.62777	0.287687
$704$	−1.79802	−0.0677655
$705$	4.45503	0.167786
$706$	47.9486	1.80457
$707$	−10.5495	−0.396756
$708$	4.74428	0.178301
$709$	16.4552	0.617987	0.308994	−	0.951064i	$-0.400008\pi$
$709$	16.4552	0.617987	0.308994	+	0.951064i	$0.400008\pi$
$710$	24.9562	0.936589
$711$	35.8936	1.34612
$712$	19.7564	0.740403
$713$	−3.80403	−0.142462
$714$	1.71071	0.0640216
$715$	2.54044	0.0950071
$716$	18.1324	0.677640
$717$	−24.0081	−0.896599
$718$	7.78548	0.290551
$719$	−27.3997	−1.02184	−0.510918	−	0.859629i	$-0.670695\pi$
$719$	−27.3997	−1.02184	−0.510918	+	0.859629i	$0.670695\pi$
$720$	−18.2695	−0.680865
$721$	−14.5220	−0.540828
$722$	29.7460	1.10703
$723$	−11.5104	−0.428078
$724$	17.0232	0.632661
$725$	−14.8266	−0.550647
$726$	−1.37187	−0.0509148
$727$	−12.8853	−0.477890	−0.238945	−	0.971033i	$-0.576802\pi$
$727$	−12.8853	−0.477890	−0.238945	+	0.971033i	$0.576802\pi$
$728$	3.78667	0.140343
$729$	1.91764	0.0710238
$730$	−30.9887	−1.14694
$731$	−1.00000	−0.0369863
$732$	−6.18925	−0.228761
$733$	44.4036	1.64008	0.820041	−	0.572304i	$-0.193950\pi$
$733$	44.4036	1.64008	0.820041	+	0.572304i	$0.193950\pi$
$734$	−28.0490	−1.03531
$735$	6.80942	0.251169
$736$	−6.90013	−0.254342
$737$	−1.04951	−0.0386593
$738$	−30.3039	−1.11550
$739$	−15.4699	−0.569068	−0.284534	−	0.958666i	$-0.591839\pi$
$739$	−15.4699	−0.569068	−0.284534	+	0.958666i	$0.591839\pi$
$740$	−8.63596	−0.317464
$741$	1.64266	0.0603445
$742$	−9.44225	−0.346636
$743$	−18.4509	−0.676899	−0.338449	−	0.940985i	$-0.609902\pi$
$743$	−18.4509	−0.676899	−0.338449	+	0.940985i	$0.609902\pi$
$744$	3.95436	0.144974
$745$	1.67165	0.0612446
$746$	−8.77433	−0.321251
$747$	−10.6456	−0.389503
$748$	0.910358	0.0332860
$749$	8.02612	0.293268
$750$	−16.1748	−0.590622
$751$	44.0506	1.60743	0.803714	−	0.595015i	$-0.202854\pi$
$751$	44.0506	1.60743	0.803714	+	0.595015i	$0.202854\pi$
$752$	17.7833	0.648489
$753$	−2.99736	−0.109230
$754$	−16.0058	−0.582898
$755$	2.17585	0.0791871
$756$	−4.88696	−0.177737
$757$	41.6434	1.51356	0.756778	−	0.653672i	$-0.226772\pi$
$757$	41.6434	1.51356	0.756778	+	0.653672i	$0.226772\pi$
$758$	24.6639	0.895834
$759$	1.15639	0.0419741
$760$	−3.61492	−0.131127
$761$	42.2549	1.53174	0.765869	−	0.642996i	$-0.222309\pi$
$761$	42.2549	1.53174	0.765869	+	0.642996i	$0.222309\pi$
$762$	−16.7161	−0.605561
$763$	13.5331	0.489931
$764$	−3.50961	−0.126973
$765$	−3.65978	−0.132320
$766$	−56.6336	−2.04625
$767$	10.5866	0.382260
$768$	14.4817	0.522564
$769$	14.2362	0.513371	0.256686	−	0.966495i	$-0.417370\pi$
$769$	14.2362	0.513371	0.256686	+	0.966495i	$0.417370\pi$
$770$	−3.30832	−0.119224
$771$	9.73335	0.350538
$772$	−2.34242	−0.0843055
$773$	−6.98985	−0.251408	−0.125704	−	0.992068i	$-0.540119\pi$
$773$	−6.98985	−0.251408	−0.125704	+	0.992068i	$0.540119\pi$
$774$	4.01474	0.144307
$775$	−6.82897	−0.245304
$776$	3.07260	0.110300
$777$	6.11687	0.219442
$778$	−33.6071	−1.20487
$779$	−9.43870	−0.338176
$780$	−1.85978	−0.0665906
$781$	−9.40662	−0.336595
$782$	−2.45322	−0.0877270
$783$	−24.7247	−0.883589
$784$	27.1813	0.970762
$785$	8.82926	0.315130
$786$	5.89836	0.210388
$787$	−30.8062	−1.09812	−0.549062	−	0.835782i	$-0.685015\pi$
$787$	−30.8062	−1.09812	−0.549062	+	0.835782i	$0.685015\pi$
$788$	−4.07138	−0.145037
$789$	22.9834	0.818232
$790$	−40.4648	−1.43967
$791$	8.22071	0.292295
$792$	4.37463	0.155446
$793$	−13.8110	−0.490442
$794$	−7.09130	−0.251661
$795$	−5.55073	−0.196864
$796$	8.99712	0.318894
$797$	−26.7852	−0.948781	−0.474391	−	0.880314i	$-0.657332\pi$
$797$	−26.7852	−0.948781	−0.474391	+	0.880314i	$0.657332\pi$
$798$	−2.13917	−0.0757258
$799$	3.56238	0.126028
$800$	−12.3871	−0.437949
$801$	−25.0112	−0.883729
$802$	−9.89796	−0.349509
$803$	11.6804	0.412194
$804$	0.768315	0.0270964
$805$	2.78868	0.0982879
$806$	−7.37210	−0.259671
$807$	−17.7635	−0.625303
$808$	−15.7263	−0.553250
$809$	48.4220	1.70243	0.851213	−	0.524820i	$-0.175867\pi$
$809$	48.4220	1.70243	0.851213	+	0.524820i	$0.175867\pi$
$810$	9.54619	0.335419
$811$	−14.5734	−0.511743	−0.255871	−	0.966711i	$-0.582362\pi$
$811$	−14.5734	−0.511743	−0.255871	+	0.966711i	$0.582362\pi$
$812$	6.51992	0.228804
$813$	1.88747	0.0661966
$814$	10.4064	0.364744
$815$	14.7881	0.518006
$816$	4.01431	0.140529
$817$	1.25046	0.0437481
$818$	−54.9498	−1.92127
$819$	−4.79385	−0.167511
$820$	10.6863	0.373180
$821$	35.3634	1.23419	0.617095	−	0.786888i	$-0.288309\pi$
$821$	35.3634	1.23419	0.617095	+	0.786888i	$0.288309\pi$
$822$	−2.20740	−0.0769919
$823$	17.2073	0.599808	0.299904	−	0.953969i	$-0.403045\pi$
$823$	17.2073	0.599808	0.299904	+	0.953969i	$0.403045\pi$
$824$	−21.6481	−0.754149
$825$	2.07594	0.0722749
$826$	−13.7865	−0.479695
$827$	36.5418	1.27068	0.635341	−	0.772232i	$-0.280860\pi$
$827$	36.5418	1.27068	0.635341	+	0.772232i	$0.280860\pi$
$828$	3.08077	0.107064
$829$	−2.16759	−0.0752834	−0.0376417	−	0.999291i	$-0.511985\pi$
$829$	−2.16759	−0.0752834	−0.0376417	+	0.999291i	$0.511985\pi$
$830$	12.0014	0.416575
$831$	24.6498	0.855094
$832$	2.93720	0.101829
$833$	5.44502	0.188659
$834$	4.54291	0.157308
$835$	25.5157	0.883006
$836$	−1.13837	−0.0393713
$837$	−11.3879	−0.393624
$838$	−52.0124	−1.79674
$839$	−10.5302	−0.363543	−0.181772	−	0.983341i	$-0.558183\pi$
$839$	−10.5302	−0.363543	−0.181772	+	0.983341i	$0.558183\pi$
$840$	−2.89888	−0.100021
$841$	3.98637	0.137461
$842$	−37.2475	−1.28363
$843$	13.6804	0.471177
$844$	10.7352	0.369520
$845$	16.0669	0.552719
$846$	−14.3020	−0.491714
$847$	1.24699	0.0428471
$848$	−22.1570	−0.760874
$849$	−13.6852	−0.469674
$850$	−4.40401	−0.151056
$851$	−8.77184	−0.300695
$852$	6.88629	0.235920
$853$	53.4605	1.83045	0.915225	−	0.402942i	$-0.132012\pi$
$853$	53.4605	1.83045	0.915225	+	0.402942i	$0.132012\pi$
$854$	17.9855	0.615451
$855$	4.57642	0.156510
$856$	11.9646	0.408943
$857$	14.8493	0.507243	0.253622	−	0.967303i	$-0.418378\pi$
$857$	14.8493	0.507243	0.253622	+	0.967303i	$0.418378\pi$
$858$	2.24104	0.0765079
$859$	−24.5646	−0.838133	−0.419067	−	0.907955i	$-0.637643\pi$
$859$	−24.5646	−0.838133	−0.419067	+	0.907955i	$0.637643\pi$
$860$	−1.41574	−0.0482763
$861$	−7.56910	−0.257954
$862$	−15.0796	−0.513614
$863$	16.9566	0.577211	0.288605	−	0.957448i	$-0.406808\pi$
$863$	16.9566	0.577211	0.288605	+	0.957448i	$0.406808\pi$
$864$	−20.6565	−0.702749
$865$	−34.7163	−1.18039
$866$	−38.1979	−1.29802
$867$	0.804154	0.0273105
$868$	3.00300	0.101928
$869$	15.2522	0.517396
$870$	12.2532	0.415424
$871$	1.71445	0.0580920
$872$	20.1740	0.683177
$873$	−3.88985	−0.131652
$874$	3.06766	0.103765
$875$	14.7025	0.497034
$876$	−8.55087	−0.288907
$877$	−8.09283	−0.273275	−0.136638	−	0.990621i	$-0.543630\pi$
$877$	−8.09283	−0.273275	−0.136638	+	0.990621i	$0.543630\pi$
$878$	14.9872	0.505795
$879$	−7.72284	−0.260485
$880$	−7.76324	−0.261699
$881$	5.25091	0.176908	0.0884539	−	0.996080i	$-0.471807\pi$
$881$	5.25091	0.176908	0.0884539	+	0.996080i	$0.471807\pi$
$882$	−21.8603	−0.736075
$883$	−33.0795	−1.11321	−0.556607	−	0.830776i	$-0.687897\pi$
$883$	−33.0795	−1.11321	−0.556607	+	0.830776i	$0.687897\pi$
$884$	−1.48713	−0.0500177
$885$	−8.10457	−0.272432
$886$	−15.0290	−0.504910
$887$	−34.9154	−1.17234	−0.586172	−	0.810186i	$-0.699366\pi$
$887$	−34.9154	−1.17234	−0.586172	+	0.810186i	$0.699366\pi$
$888$	9.11850	0.305997
$889$	15.1945	0.509607
$890$	28.1965	0.945150
$891$	−3.59820	−0.120544
$892$	−15.3720	−0.514691
$893$	−4.45462	−0.149068
$894$	1.47464	0.0493194
$895$	−30.9753	−1.03539
$896$	−15.7920	−0.527575
$897$	−1.88904	−0.0630731
$898$	−23.4840	−0.783672
$899$	15.1931	0.506720
$900$	5.53059	0.184353
$901$	−4.43853	−0.147869
$902$	−12.8770	−0.428758
$903$	1.00277	0.0333702
$904$	12.2547	0.407586
$905$	−29.0803	−0.966663
$906$	1.91941	0.0637683
$907$	30.0043	0.996277	0.498139	−	0.867097i	$-0.334017\pi$
$907$	30.0043	0.996277	0.498139	+	0.867097i	$0.334017\pi$
$908$	22.6587	0.751957
$909$	19.9092	0.660347
$910$	5.40437	0.179153
$911$	1.63418	0.0541429	0.0270715	−	0.999634i	$-0.491382\pi$
$911$	1.63418	0.0541429	0.0270715	+	0.999634i	$0.491382\pi$
$912$	−5.01974	−0.166220
$913$	−4.52363	−0.149710
$914$	30.0338	0.993430
$915$	10.5730	0.349532
$916$	−20.3928	−0.673798
$917$	−5.36145	−0.177051
$918$	−7.34407	−0.242390
$919$	−20.2999	−0.669632	−0.334816	−	0.942284i	$-0.608674\pi$
$919$	−20.2999	−0.669632	−0.334816	+	0.942284i	$0.608674\pi$
$920$	4.15712	0.137056
$921$	27.3573	0.901454
$922$	−68.7075	−2.26276
$923$	15.3664	0.505790
$924$	−0.912881	−0.0300316
$925$	−15.7472	−0.517763
$926$	−47.0493	−1.54613
$927$	27.4061	0.900135
$928$	27.5588	0.904663
$929$	9.79492	0.321361	0.160680	−	0.987006i	$-0.448631\pi$
$929$	9.79492	0.321361	0.160680	+	0.987006i	$0.448631\pi$
$930$	5.64370	0.185064
$931$	−6.80878	−0.223149
$932$	4.26490	0.139701
$933$	−25.9386	−0.849192
$934$	−4.82566	−0.157901
$935$	−1.55515	−0.0508587
$936$	−7.14626	−0.233583
$937$	27.5425	0.899774	0.449887	−	0.893086i	$-0.351464\pi$
$937$	27.5425	0.899774	0.449887	+	0.893086i	$0.351464\pi$
$938$	−2.23267	−0.0728992
$939$	22.4296	0.731963
$940$	5.04340	0.164498
$941$	46.1347	1.50395	0.751974	−	0.659193i	$-0.229102\pi$
$941$	46.1347	1.50395	0.751974	+	0.659193i	$0.229102\pi$
$942$	7.78871	0.253770
$943$	10.8544	0.353468
$944$	−32.3512	−1.05294
$945$	8.34830	0.271570
$946$	1.70598	0.0554661
$947$	59.0440	1.91867	0.959336	−	0.282267i	$-0.0910864\pi$
$947$	59.0440	1.91867	0.959336	+	0.282267i	$0.0910864\pi$
$948$	−11.1657	−0.362644
$949$	−19.0808	−0.619389
$950$	5.50704	0.178672
$951$	−14.5924	−0.473193
$952$	−2.31804	−0.0751279
$953$	9.83928	0.318726	0.159363	−	0.987220i	$-0.449056\pi$
$953$	9.83928	0.318726	0.159363	+	0.987220i	$0.449056\pi$
$954$	17.8195	0.576929
$955$	5.99541	0.194007
$956$	−27.1788	−0.879027
$957$	−4.61856	−0.149297
$958$	44.1999	1.42803
$959$	2.00647	0.0647922
$960$	−2.24857	−0.0725722
$961$	−24.0022	−0.774265
$962$	−16.9996	−0.548088
$963$	−15.1470	−0.488105
$964$	−13.0306	−0.419688
$965$	4.00151	0.128813
$966$	2.46002	0.0791499
$967$	43.0829	1.38545	0.692726	−	0.721201i	$-0.256410\pi$
$967$	43.0829	1.38545	0.692726	+	0.721201i	$0.256410\pi$
$968$	1.85890	0.0597475
$969$	−1.00556	−0.0323033
$970$	4.38524	0.140802
$971$	−47.9216	−1.53788	−0.768939	−	0.639322i	$-0.779215\pi$
$971$	−47.9216	−1.53788	−0.768939	+	0.639322i	$0.779215\pi$
$972$	14.3911	0.461596
$973$	−4.12938	−0.132382
$974$	−1.20623	−0.0386501
$975$	−3.39119	−0.108605
$976$	42.2044	1.35093
$977$	9.15662	0.292946	0.146473	−	0.989215i	$-0.453208\pi$
$977$	9.15662	0.292946	0.146473	+	0.989215i	$0.453208\pi$
$978$	13.0453	0.417143
$979$	−10.6280	−0.339672
$980$	7.70873	0.246246
$981$	−25.5399	−0.815425
$982$	−28.7197	−0.916483
$983$	37.0199	1.18075	0.590376	−	0.807129i	$-0.298980\pi$
$983$	37.0199	1.18075	0.590376	+	0.807129i	$0.298980\pi$
$984$	−11.2834	−0.359700
$985$	6.95506	0.221607
$986$	9.79807	0.312034
$987$	−3.57225	−0.113706
$988$	1.85960	0.0591618
$989$	−1.43802	−0.0457262
$990$	6.24351	0.198432
$991$	28.6420	0.909844	0.454922	−	0.890531i	$-0.349667\pi$
$991$	28.6420	0.909844	0.454922	+	0.890531i	$0.349667\pi$
$992$	12.6933	0.403012
$993$	−2.13790	−0.0678443
$994$	−20.0110	−0.634712
$995$	−15.3696	−0.487249
$996$	3.31161	0.104932
$997$	−40.0369	−1.26798	−0.633991	−	0.773340i	$-0.718584\pi$
$997$	−40.0369	−1.26798	−0.633991	+	0.773340i	$0.718584\pi$
$998$	−47.5780	−1.50606
$999$	−26.2598	−0.830822

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.e.1.14	✓	66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.e.1.14	✓	66	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8041 = 11 \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8041.a (trivial)

\(f(q)\)	\(=\)	\(q-1.70598 q^{2} +0.804154 q^{3} +0.910358 q^{4} -1.55515 q^{5} -1.37187 q^{6} +1.24699 q^{7} +1.85890 q^{8} -2.35334 q^{9} +O(q^{10})\)	\(q-1.70598 q^{2} +0.804154 q^{3} +0.910358 q^{4} -1.55515 q^{5} -1.37187 q^{6} +1.24699 q^{7} +1.85890 q^{8} -2.35334 q^{9} +2.65305 q^{10} -1.00000 q^{11} +0.732068 q^{12} +1.63357 q^{13} -2.12734 q^{14} -1.25058 q^{15} -4.99196 q^{16} -1.00000 q^{17} +4.01474 q^{18} +1.25046 q^{19} -1.41574 q^{20} +1.00277 q^{21} +1.70598 q^{22} -1.43802 q^{23} +1.49485 q^{24} -2.58152 q^{25} -2.78683 q^{26} -4.30491 q^{27} +1.13521 q^{28} +5.74338 q^{29} +2.13346 q^{30} +2.64533 q^{31} +4.79837 q^{32} -0.804154 q^{33} +1.70598 q^{34} -1.93925 q^{35} -2.14238 q^{36} +6.09996 q^{37} -2.13326 q^{38} +1.31364 q^{39} -2.89087 q^{40} -7.54817 q^{41} -1.71071 q^{42} +1.00000 q^{43} -0.910358 q^{44} +3.65978 q^{45} +2.45322 q^{46} -3.56238 q^{47} -4.01431 q^{48} -5.44502 q^{49} +4.40401 q^{50} -0.804154 q^{51} +1.48713 q^{52} +4.43853 q^{53} +7.34407 q^{54} +1.55515 q^{55} +2.31804 q^{56} +1.00556 q^{57} -9.79807 q^{58} +6.48066 q^{59} -1.13847 q^{60} -8.45447 q^{61} -4.51288 q^{62} -2.93459 q^{63} +1.79802 q^{64} -2.54044 q^{65} +1.37187 q^{66} +1.04951 q^{67} -0.910358 q^{68} -1.15639 q^{69} +3.30832 q^{70} +9.40662 q^{71} -4.37463 q^{72} -11.6804 q^{73} -10.4064 q^{74} -2.07594 q^{75} +1.13837 q^{76} -1.24699 q^{77} -2.24104 q^{78} -15.2522 q^{79} +7.76324 q^{80} +3.59820 q^{81} +12.8770 q^{82} +4.52363 q^{83} +0.912881 q^{84} +1.55515 q^{85} -1.70598 q^{86} +4.61856 q^{87} -1.85890 q^{88} +10.6280 q^{89} -6.24351 q^{90} +2.03705 q^{91} -1.30911 q^{92} +2.12725 q^{93} +6.07734 q^{94} -1.94465 q^{95} +3.85863 q^{96} +1.65291 q^{97} +9.28907 q^{98} +2.35334 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 7 q^{2} + 3 q^{3} + 61 q^{4} + 4 q^{5} + 10 q^{6} + 14 q^{7} + 21 q^{8} + 65 q^{9}+O(q^{10}) \) 66 * q + 7 * q^2 + 3 * q^3 + 61 * q^4 + 4 * q^5 + 10 * q^6 + 14 * q^7 + 21 * q^8 + 65 * q^9	\( 66 q + 7 q^{2} + 3 q^{3} + 61 q^{4} + 4 q^{5} + 10 q^{6} + 14 q^{7} + 21 q^{8} + 65 q^{9} + 13 q^{10} - 66 q^{11} + 9 q^{12} - 12 q^{13} + 25 q^{14} + 13 q^{15} + 47 q^{16} - 66 q^{17} + 37 q^{18} + 19 q^{20} + 26 q^{21} - 7 q^{22} + 47 q^{23} + 15 q^{24} + 52 q^{25} + 16 q^{26} + 9 q^{27} + 3 q^{28} + 57 q^{29} + 2 q^{30} + 31 q^{31} + 39 q^{32} - 3 q^{33} - 7 q^{34} + 36 q^{35} + 39 q^{36} - 14 q^{37} + 18 q^{38} + 71 q^{39} + 29 q^{40} + 62 q^{41} - 3 q^{42} + 66 q^{43} - 61 q^{44} - 2 q^{45} + 19 q^{46} + 32 q^{47} + 26 q^{48} + 42 q^{49} + 10 q^{50} - 3 q^{51} - 7 q^{52} + 33 q^{53} + 100 q^{54} - 4 q^{55} + 61 q^{56} + 35 q^{57} - 16 q^{58} + 59 q^{59} + 50 q^{60} + 26 q^{61} + 29 q^{62} + 62 q^{63} + 29 q^{64} + 55 q^{65} - 10 q^{66} + 5 q^{67} - 61 q^{68} - 36 q^{69} - 35 q^{70} + 128 q^{71} + 87 q^{72} + 23 q^{73} + 64 q^{74} - 11 q^{75} + 74 q^{76} - 14 q^{77} + 45 q^{78} + 39 q^{79} + 95 q^{80} + 54 q^{81} - 6 q^{82} + 48 q^{83} + 38 q^{84} - 4 q^{85} + 7 q^{86} + 14 q^{87} - 21 q^{88} + 28 q^{89} + 135 q^{90} - 18 q^{91} + 108 q^{92} - 9 q^{93} + 37 q^{94} + 149 q^{95} + 104 q^{96} + 19 q^{97} + 30 q^{98} - 65 q^{99}+O(q^{100}) \) 66 * q + 7 * q^2 + 3 * q^3 + 61 * q^4 + 4 * q^5 + 10 * q^6 + 14 * q^7 + 21 * q^8 + 65 * q^9 + 13 * q^10 - 66 * q^11 + 9 * q^12 - 12 * q^13 + 25 * q^14 + 13 * q^15 + 47 * q^16 - 66 * q^17 + 37 * q^18 + 19 * q^20 + 26 * q^21 - 7 * q^22 + 47 * q^23 + 15 * q^24 + 52 * q^25 + 16 * q^26 + 9 * q^27 + 3 * q^28 + 57 * q^29 + 2 * q^30 + 31 * q^31 + 39 * q^32 - 3 * q^33 - 7 * q^34 + 36 * q^35 + 39 * q^36 - 14 * q^37 + 18 * q^38 + 71 * q^39 + 29 * q^40 + 62 * q^41 - 3 * q^42 + 66 * q^43 - 61 * q^44 - 2 * q^45 + 19 * q^46 + 32 * q^47 + 26 * q^48 + 42 * q^49 + 10 * q^50 - 3 * q^51 - 7 * q^52 + 33 * q^53 + 100 * q^54 - 4 * q^55 + 61 * q^56 + 35 * q^57 - 16 * q^58 + 59 * q^59 + 50 * q^60 + 26 * q^61 + 29 * q^62 + 62 * q^63 + 29 * q^64 + 55 * q^65 - 10 * q^66 + 5 * q^67 - 61 * q^68 - 36 * q^69 - 35 * q^70 + 128 * q^71 + 87 * q^72 + 23 * q^73 + 64 * q^74 - 11 * q^75 + 74 * q^76 - 14 * q^77 + 45 * q^78 + 39 * q^79 + 95 * q^80 + 54 * q^81 - 6 * q^82 + 48 * q^83 + 38 * q^84 - 4 * q^85 + 7 * q^86 + 14 * q^87 - 21 * q^88 + 28 * q^89 + 135 * q^90 - 18 * q^91 + 108 * q^92 - 9 * q^93 + 37 * q^94 + 149 * q^95 + 104 * q^96 + 19 * q^97 + 30 * q^98 - 65 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more