LMFDB - Embedded newform 6008.2.a.e.1.39

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, 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("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.39
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.13242 q^{3} -1.12247 q^{5} -4.02550 q^{7} +1.54720 q^{9} +O(q^{10})$	$q+2.13242 q^{3} -1.12247 q^{5} -4.02550 q^{7} +1.54720 q^{9} -5.57155 q^{11} +0.335534 q^{13} -2.39358 q^{15} +4.79184 q^{17} +1.73171 q^{19} -8.58404 q^{21} -3.07533 q^{23} -3.74005 q^{25} -3.09798 q^{27} +6.66349 q^{29} -3.51015 q^{31} -11.8809 q^{33} +4.51852 q^{35} +5.13640 q^{37} +0.715498 q^{39} +4.64579 q^{41} -9.81357 q^{43} -1.73669 q^{45} +11.9133 q^{47} +9.20464 q^{49} +10.2182 q^{51} +11.4349 q^{53} +6.25393 q^{55} +3.69273 q^{57} +2.35959 q^{59} +9.55165 q^{61} -6.22824 q^{63} -0.376628 q^{65} -2.78716 q^{67} -6.55787 q^{69} +7.12313 q^{71} +6.65412 q^{73} -7.97534 q^{75} +22.4283 q^{77} -8.62639 q^{79} -11.2478 q^{81} -1.96497 q^{83} -5.37873 q^{85} +14.2093 q^{87} +14.7566 q^{89} -1.35069 q^{91} -7.48510 q^{93} -1.94380 q^{95} +5.97150 q^{97} -8.62029 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * 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 $

$2$	0	0
$2$	0	0
$3$	2.13242	1.23115	0.615576	−	0.788078i	$-0.288924\pi$
$3$	2.13242	1.23115	0.615576	+	0.788078i	$0.288924\pi$
$4$	0	0
$5$	−1.12247	−0.501986	−0.250993	−	0.967989i	$-0.580757\pi$
$5$	−1.12247	−0.501986	−0.250993	+	0.967989i	$0.580757\pi$
$6$	0	0
$7$	−4.02550	−1.52150	−0.760748	−	0.649048i	$-0.775168\pi$
$7$	−4.02550	−1.52150	−0.760748	+	0.649048i	$0.775168\pi$
$8$	0	0
$9$	1.54720	0.515733
$10$	0	0
$11$	−5.57155	−1.67989	−0.839943	−	0.542675i	$-0.817412\pi$
$11$	−5.57155	−1.67989	−0.839943	+	0.542675i	$0.817412\pi$
$12$	0	0
$13$	0.335534	0.0930603	0.0465302	−	0.998917i	$-0.485184\pi$
$13$	0.335534	0.0930603	0.0465302	+	0.998917i	$0.485184\pi$
$14$	0	0
$15$	−2.39358	−0.618021
$16$	0	0
$17$	4.79184	1.16219	0.581097	−	0.813835i	$-0.302624\pi$
$17$	4.79184	1.16219	0.581097	+	0.813835i	$0.302624\pi$
$18$	0	0
$19$	1.73171	0.397282	0.198641	−	0.980072i	$-0.436347\pi$
$19$	1.73171	0.397282	0.198641	+	0.980072i	$0.436347\pi$
$20$	0	0
$21$	−8.58404	−1.87319
$22$	0	0
$23$	−3.07533	−0.641250	−0.320625	−	0.947206i	$-0.603893\pi$
$23$	−3.07533	−0.641250	−0.320625	+	0.947206i	$0.603893\pi$
$24$	0	0
$25$	−3.74005	−0.748010
$26$	0	0
$27$	−3.09798	−0.596206
$28$	0	0
$29$	6.66349	1.23738	0.618689	−	0.785636i	$-0.287664\pi$
$29$	6.66349	1.23738	0.618689	+	0.785636i	$0.287664\pi$
$30$	0	0
$31$	−3.51015	−0.630442	−0.315221	−	0.949018i	$-0.602079\pi$
$31$	−3.51015	−0.630442	−0.315221	+	0.949018i	$0.602079\pi$
$32$	0	0
$33$	−11.8809	−2.06819
$34$	0	0
$35$	4.51852	0.763769
$36$	0	0
$37$	5.13640	0.844418	0.422209	−	0.906498i	$-0.361255\pi$
$37$	5.13640	0.844418	0.422209	+	0.906498i	$0.361255\pi$
$38$	0	0
$39$	0.715498	0.114571
$40$	0	0
$41$	4.64579	0.725551	0.362775	−	0.931877i	$-0.381829\pi$
$41$	4.64579	0.725551	0.362775	+	0.931877i	$0.381829\pi$
$42$	0	0
$43$	−9.81357	−1.49656	−0.748278	−	0.663386i	$-0.769119\pi$
$43$	−9.81357	−1.49656	−0.748278	+	0.663386i	$0.769119\pi$
$44$	0	0
$45$	−1.73669	−0.258891
$46$	0	0
$47$	11.9133	1.73773	0.868864	−	0.495051i	$-0.164851\pi$
$47$	11.9133	1.73773	0.868864	+	0.495051i	$0.164851\pi$
$48$	0	0
$49$	9.20464	1.31495
$50$	0	0
$51$	10.2182	1.43084
$52$	0	0
$53$	11.4349	1.57070	0.785351	−	0.619050i	$-0.212482\pi$
$53$	11.4349	1.57070	0.785351	+	0.619050i	$0.212482\pi$
$54$	0	0
$55$	6.25393	0.843279
$56$	0	0
$57$	3.69273	0.489114
$58$	0	0
$59$	2.35959	0.307192	0.153596	−	0.988134i	$-0.450915\pi$
$59$	2.35959	0.307192	0.153596	+	0.988134i	$0.450915\pi$
$60$	0	0
$61$	9.55165	1.22296	0.611482	−	0.791259i	$-0.290574\pi$
$61$	9.55165	1.22296	0.611482	+	0.791259i	$0.290574\pi$
$62$	0	0
$63$	−6.22824	−0.784685
$64$	0	0
$65$	−0.376628	−0.0467150
$66$	0	0
$67$	−2.78716	−0.340506	−0.170253	−	0.985400i	$-0.554458\pi$
$67$	−2.78716	−0.340506	−0.170253	+	0.985400i	$0.554458\pi$
$68$	0	0
$69$	−6.55787	−0.789475
$70$	0	0
$71$	7.12313	0.845361	0.422680	−	0.906279i	$-0.361089\pi$
$71$	7.12313	0.845361	0.422680	+	0.906279i	$0.361089\pi$
$72$	0	0
$73$	6.65412	0.778806	0.389403	−	0.921067i	$-0.372681\pi$
$73$	6.65412	0.778806	0.389403	+	0.921067i	$0.372681\pi$
$74$	0	0
$75$	−7.97534	−0.920913
$76$	0	0
$77$	22.4283	2.55594
$78$	0	0
$79$	−8.62639	−0.970545	−0.485273	−	0.874363i	$-0.661280\pi$
$79$	−8.62639	−0.970545	−0.485273	+	0.874363i	$0.661280\pi$
$80$	0	0
$81$	−11.2478	−1.24975
$82$	0	0
$83$	−1.96497	−0.215684	−0.107842	−	0.994168i	$-0.534394\pi$
$83$	−1.96497	−0.215684	−0.107842	+	0.994168i	$0.534394\pi$
$84$	0	0
$85$	−5.37873	−0.583405
$86$	0	0
$87$	14.2093	1.52340
$88$	0	0
$89$	14.7566	1.56420	0.782098	−	0.623156i	$-0.214150\pi$
$89$	14.7566	1.56420	0.782098	+	0.623156i	$0.214150\pi$
$90$	0	0
$91$	−1.35069	−0.141591
$92$	0	0
$93$	−7.48510	−0.776169
$94$	0	0
$95$	−1.94380	−0.199430
$96$	0	0
$97$	5.97150	0.606313	0.303157	−	0.952941i	$-0.401959\pi$
$97$	5.97150	0.606313	0.303157	+	0.952941i	$0.401959\pi$
$98$	0	0
$99$	−8.62029	−0.866372
$100$	0	0
$101$	−3.40272	−0.338583	−0.169292	−	0.985566i	$-0.554148\pi$
$101$	−3.40272	−0.338583	−0.169292	+	0.985566i	$0.554148\pi$
$102$	0	0
$103$	1.65704	0.163273	0.0816363	−	0.996662i	$-0.473985\pi$
$103$	1.65704	0.163273	0.0816363	+	0.996662i	$0.473985\pi$
$104$	0	0
$105$	9.63537	0.940315
$106$	0	0
$107$	−3.63844	−0.351741	−0.175870	−	0.984413i	$-0.556274\pi$
$107$	−3.63844	−0.351741	−0.175870	+	0.984413i	$0.556274\pi$
$108$	0	0
$109$	−0.535565	−0.0512978	−0.0256489	−	0.999671i	$-0.508165\pi$
$109$	−0.535565	−0.0512978	−0.0256489	+	0.999671i	$0.508165\pi$
$110$	0	0
$111$	10.9529	1.03961
$112$	0	0
$113$	−16.9182	−1.59153	−0.795765	−	0.605606i	$-0.792931\pi$
$113$	−16.9182	−1.59153	−0.795765	+	0.605606i	$0.792931\pi$
$114$	0	0
$115$	3.45198	0.321898
$116$	0	0
$117$	0.519137	0.0479943
$118$	0	0
$119$	−19.2896	−1.76827
$120$	0	0
$121$	20.0422	1.82202
$122$	0	0
$123$	9.90676	0.893262
$124$	0	0
$125$	9.81049	0.877477
$126$	0	0
$127$	9.05841	0.803804	0.401902	−	0.915683i	$-0.368349\pi$
$127$	9.05841	0.803804	0.401902	+	0.915683i	$0.368349\pi$
$128$	0	0
$129$	−20.9266	−1.84249
$130$	0	0
$131$	10.6997	0.934835	0.467418	−	0.884037i	$-0.345184\pi$
$131$	10.6997	0.934835	0.467418	+	0.884037i	$0.345184\pi$
$132$	0	0
$133$	−6.97101	−0.604463
$134$	0	0
$135$	3.47740	0.299287
$136$	0	0
$137$	−6.52914	−0.557822	−0.278911	−	0.960317i	$-0.589973\pi$
$137$	−6.52914	−0.557822	−0.278911	+	0.960317i	$0.589973\pi$
$138$	0	0
$139$	−13.6873	−1.16094	−0.580472	−	0.814280i	$-0.697132\pi$
$139$	−13.6873	−1.16094	−0.580472	+	0.814280i	$0.697132\pi$
$140$	0	0
$141$	25.4040	2.13941
$142$	0	0
$143$	−1.86944	−0.156331
$144$	0	0
$145$	−7.47960	−0.621147
$146$	0	0
$147$	19.6281	1.61890
$148$	0	0
$149$	−15.7650	−1.29152	−0.645758	−	0.763542i	$-0.723459\pi$
$149$	−15.7650	−1.29152	−0.645758	+	0.763542i	$0.723459\pi$
$150$	0	0
$151$	8.93828	0.727387	0.363694	−	0.931519i	$-0.381516\pi$
$151$	8.93828	0.727387	0.363694	+	0.931519i	$0.381516\pi$
$152$	0	0
$153$	7.41393	0.599381
$154$	0	0
$155$	3.94006	0.316473
$156$	0	0
$157$	0.790479	0.0630871	0.0315435	−	0.999502i	$-0.489958\pi$
$157$	0.790479	0.0630871	0.0315435	+	0.999502i	$0.489958\pi$
$158$	0	0
$159$	24.3839	1.93377
$160$	0	0
$161$	12.3797	0.975659
$162$	0	0
$163$	23.8021	1.86433	0.932164	−	0.362036i	$-0.117918\pi$
$163$	23.8021	1.86433	0.932164	+	0.362036i	$0.117918\pi$
$164$	0	0
$165$	13.3360	1.03820
$166$	0	0
$167$	−12.9128	−0.999223	−0.499611	−	0.866250i	$-0.666524\pi$
$167$	−12.9128	−0.999223	−0.499611	+	0.866250i	$0.666524\pi$
$168$	0	0
$169$	−12.8874	−0.991340
$170$	0	0
$171$	2.67930	0.204891
$172$	0	0
$173$	13.0712	0.993782	0.496891	−	0.867813i	$-0.334475\pi$
$173$	13.0712	0.993782	0.496891	+	0.867813i	$0.334475\pi$
$174$	0	0
$175$	15.0556	1.13809
$176$	0	0
$177$	5.03162	0.378200
$178$	0	0
$179$	9.20320	0.687880	0.343940	−	0.938992i	$-0.388238\pi$
$179$	9.20320	0.687880	0.343940	+	0.938992i	$0.388238\pi$
$180$	0	0
$181$	0.613750	0.0456197	0.0228098	−	0.999740i	$-0.492739\pi$
$181$	0.613750	0.0456197	0.0228098	+	0.999740i	$0.492739\pi$
$182$	0	0
$183$	20.3681	1.50565
$184$	0	0
$185$	−5.76548	−0.423886
$186$	0	0
$187$	−26.6980	−1.95235
$188$	0	0
$189$	12.4709	0.907125
$190$	0	0
$191$	6.31450	0.456901	0.228451	−	0.973556i	$-0.426634\pi$
$191$	6.31450	0.456901	0.228451	+	0.973556i	$0.426634\pi$
$192$	0	0
$193$	10.8091	0.778052	0.389026	−	0.921227i	$-0.372812\pi$
$193$	10.8091	0.778052	0.389026	+	0.921227i	$0.372812\pi$
$194$	0	0
$195$	−0.803128	−0.0575132
$196$	0	0
$197$	20.9045	1.48938	0.744692	−	0.667409i	$-0.232597\pi$
$197$	20.9045	1.48938	0.744692	+	0.667409i	$0.232597\pi$
$198$	0	0
$199$	14.3645	1.01828	0.509138	−	0.860685i	$-0.329964\pi$
$199$	14.3645	1.01828	0.509138	+	0.860685i	$0.329964\pi$
$200$	0	0
$201$	−5.94338	−0.419214
$202$	0	0
$203$	−26.8239	−1.88267
$204$	0	0
$205$	−5.21478	−0.364216
$206$	0	0
$207$	−4.75814	−0.330714
$208$	0	0
$209$	−9.64833	−0.667389
$210$	0	0
$211$	16.9552	1.16724	0.583622	−	0.812025i	$-0.301635\pi$
$211$	16.9552	1.16724	0.583622	+	0.812025i	$0.301635\pi$
$212$	0	0
$213$	15.1895	1.04077
$214$	0	0
$215$	11.0155	0.751250
$216$	0	0
$217$	14.1301	0.959214
$218$	0	0
$219$	14.1894	0.958828
$220$	0	0
$221$	1.60783	0.108154
$222$	0	0
$223$	22.9004	1.53352	0.766762	−	0.641932i	$-0.221867\pi$
$223$	22.9004	1.53352	0.766762	+	0.641932i	$0.221867\pi$
$224$	0	0
$225$	−5.78660	−0.385773
$226$	0	0
$227$	−19.1925	−1.27385	−0.636924	−	0.770926i	$-0.719794\pi$
$227$	−19.1925	−1.27385	−0.636924	+	0.770926i	$0.719794\pi$
$228$	0	0
$229$	−4.16542	−0.275258	−0.137629	−	0.990484i	$-0.543948\pi$
$229$	−4.16542	−0.275258	−0.137629	+	0.990484i	$0.543948\pi$
$230$	0	0
$231$	47.8264	3.14675
$232$	0	0
$233$	−24.3789	−1.59711	−0.798557	−	0.601920i	$-0.794403\pi$
$233$	−24.3789	−1.59711	−0.798557	+	0.601920i	$0.794403\pi$
$234$	0	0
$235$	−13.3723	−0.872315
$236$	0	0
$237$	−18.3951	−1.19489
$238$	0	0
$239$	8.48871	0.549089	0.274545	−	0.961574i	$-0.411473\pi$
$239$	8.48871	0.549089	0.274545	+	0.961574i	$0.411473\pi$
$240$	0	0
$241$	−14.4061	−0.927978	−0.463989	−	0.885841i	$-0.653582\pi$
$241$	−14.4061	−0.927978	−0.463989	+	0.885841i	$0.653582\pi$
$242$	0	0
$243$	−14.6910	−0.942428
$244$	0	0
$245$	−10.3320	−0.660086
$246$	0	0
$247$	0.581048	0.0369712
$248$	0	0
$249$	−4.19014	−0.265539
$250$	0	0
$251$	−12.3277	−0.778118	−0.389059	−	0.921213i	$-0.627200\pi$
$251$	−12.3277	−0.778118	−0.389059	+	0.921213i	$0.627200\pi$
$252$	0	0
$253$	17.1343	1.07723
$254$	0	0
$255$	−11.4697	−0.718259
$256$	0	0
$257$	−21.1730	−1.32073	−0.660366	−	0.750944i	$-0.729599\pi$
$257$	−21.1730	−1.32073	−0.660366	+	0.750944i	$0.729599\pi$
$258$	0	0
$259$	−20.6766	−1.28478
$260$	0	0
$261$	10.3097	0.638157
$262$	0	0
$263$	17.0142	1.04914	0.524570	−	0.851367i	$-0.324226\pi$
$263$	17.0142	1.04914	0.524570	+	0.851367i	$0.324226\pi$
$264$	0	0
$265$	−12.8354	−0.788471
$266$	0	0
$267$	31.4672	1.92576
$268$	0	0
$269$	−13.9957	−0.853334	−0.426667	−	0.904409i	$-0.640312\pi$
$269$	−13.9957	−0.853334	−0.426667	+	0.904409i	$0.640312\pi$
$270$	0	0
$271$	−11.1835	−0.679351	−0.339675	−	0.940543i	$-0.610317\pi$
$271$	−11.1835	−0.679351	−0.339675	+	0.940543i	$0.610317\pi$
$272$	0	0
$273$	−2.88023	−0.174320
$274$	0	0
$275$	20.8379	1.25657
$276$	0	0
$277$	12.3565	0.742431	0.371216	−	0.928547i	$-0.378941\pi$
$277$	12.3565	0.742431	0.371216	+	0.928547i	$0.378941\pi$
$278$	0	0
$279$	−5.43090	−0.325140
$280$	0	0
$281$	7.82560	0.466836	0.233418	−	0.972376i	$-0.425009\pi$
$281$	7.82560	0.466836	0.233418	+	0.972376i	$0.425009\pi$
$282$	0	0
$283$	7.91901	0.470736	0.235368	−	0.971906i	$-0.424370\pi$
$283$	7.91901	0.470736	0.235368	+	0.971906i	$0.424370\pi$
$284$	0	0
$285$	−4.14500	−0.245529
$286$	0	0
$287$	−18.7016	−1.10392
$288$	0	0
$289$	5.96178	0.350693
$290$	0	0
$291$	12.7337	0.746463
$292$	0	0
$293$	5.56530	0.325128	0.162564	−	0.986698i	$-0.448024\pi$
$293$	5.56530	0.325128	0.162564	+	0.986698i	$0.448024\pi$
$294$	0	0
$295$	−2.64858	−0.154206
$296$	0	0
$297$	17.2605	1.00156
$298$	0	0
$299$	−1.03188	−0.0596749
$300$	0	0
$301$	39.5045	2.27700
$302$	0	0
$303$	−7.25601	−0.416847
$304$	0	0
$305$	−10.7215	−0.613910
$306$	0	0
$307$	1.54406	0.0881243	0.0440622	−	0.999029i	$-0.485970\pi$
$307$	1.54406	0.0881243	0.0440622	+	0.999029i	$0.485970\pi$
$308$	0	0
$309$	3.53349	0.201013
$310$	0	0
$311$	−25.6694	−1.45558	−0.727789	−	0.685801i	$-0.759452\pi$
$311$	−25.6694	−1.45558	−0.727789	+	0.685801i	$0.759452\pi$
$312$	0	0
$313$	−19.0928	−1.07919	−0.539596	−	0.841924i	$-0.681423\pi$
$313$	−19.0928	−1.07919	−0.539596	+	0.841924i	$0.681423\pi$
$314$	0	0
$315$	6.99105	0.393901
$316$	0	0
$317$	−19.9361	−1.11972	−0.559862	−	0.828586i	$-0.689146\pi$
$317$	−19.9361	−1.11972	−0.559862	+	0.828586i	$0.689146\pi$
$318$	0	0
$319$	−37.1260	−2.07865
$320$	0	0
$321$	−7.75866	−0.433046
$322$	0	0
$323$	8.29810	0.461719
$324$	0	0
$325$	−1.25491	−0.0696101
$326$	0	0
$327$	−1.14205	−0.0631553
$328$	0	0
$329$	−47.9568	−2.64394
$330$	0	0
$331$	32.2130	1.77059	0.885293	−	0.465033i	$-0.153958\pi$
$331$	32.2130	1.77059	0.885293	+	0.465033i	$0.153958\pi$
$332$	0	0
$333$	7.94702	0.435494
$334$	0	0
$335$	3.12851	0.170929
$336$	0	0
$337$	30.0342	1.63606	0.818032	−	0.575172i	$-0.195065\pi$
$337$	30.0342	1.63606	0.818032	+	0.575172i	$0.195065\pi$
$338$	0	0
$339$	−36.0766	−1.95941
$340$	0	0
$341$	19.5570	1.05907
$342$	0	0
$343$	−8.87477	−0.479193
$344$	0	0
$345$	7.36105	0.396306
$346$	0	0
$347$	31.5880	1.69573	0.847866	−	0.530211i	$-0.177887\pi$
$347$	31.5880	1.69573	0.847866	+	0.530211i	$0.177887\pi$
$348$	0	0
$349$	2.95362	0.158103	0.0790517	−	0.996871i	$-0.474811\pi$
$349$	2.95362	0.158103	0.0790517	+	0.996871i	$0.474811\pi$
$350$	0	0
$351$	−1.03948	−0.0554831
$352$	0	0
$353$	16.8619	0.897470	0.448735	−	0.893665i	$-0.351875\pi$
$353$	16.8619	0.897470	0.448735	+	0.893665i	$0.351875\pi$
$354$	0	0
$355$	−7.99554	−0.424359
$356$	0	0
$357$	−41.1334	−2.17701
$358$	0	0
$359$	15.0461	0.794105	0.397052	−	0.917796i	$-0.370033\pi$
$359$	15.0461	0.794105	0.397052	+	0.917796i	$0.370033\pi$
$360$	0	0
$361$	−16.0012	−0.842167
$362$	0	0
$363$	42.7383	2.24318
$364$	0	0
$365$	−7.46909	−0.390950
$366$	0	0
$367$	−5.35745	−0.279657	−0.139828	−	0.990176i	$-0.544655\pi$
$367$	−5.35745	−0.279657	−0.139828	+	0.990176i	$0.544655\pi$
$368$	0	0
$369$	7.18796	0.374190
$370$	0	0
$371$	−46.0311	−2.38982
$372$	0	0
$373$	−25.7958	−1.33565	−0.667827	−	0.744316i	$-0.732776\pi$
$373$	−25.7958	−1.33565	−0.667827	+	0.744316i	$0.732776\pi$
$374$	0	0
$375$	20.9200	1.08031
$376$	0	0
$377$	2.23582	0.115151
$378$	0	0
$379$	5.57137	0.286182	0.143091	−	0.989710i	$-0.454296\pi$
$379$	5.57137	0.286182	0.143091	+	0.989710i	$0.454296\pi$
$380$	0	0
$381$	19.3163	0.989604
$382$	0	0
$383$	35.7960	1.82909	0.914545	−	0.404485i	$-0.132549\pi$
$383$	35.7960	1.82909	0.914545	+	0.404485i	$0.132549\pi$
$384$	0	0
$385$	−25.1752	−1.28305
$386$	0	0
$387$	−15.1835	−0.771823
$388$	0	0
$389$	11.5271	0.584448	0.292224	−	0.956350i	$-0.405605\pi$
$389$	11.5271	0.584448	0.292224	+	0.956350i	$0.405605\pi$
$390$	0	0
$391$	−14.7365	−0.745256
$392$	0	0
$393$	22.8162	1.15092
$394$	0	0
$395$	9.68291	0.487200
$396$	0	0
$397$	23.3988	1.17435	0.587175	−	0.809460i	$-0.300240\pi$
$397$	23.3988	1.17435	0.587175	+	0.809460i	$0.300240\pi$
$398$	0	0
$399$	−14.8651	−0.744185
$400$	0	0
$401$	−9.59029	−0.478916	−0.239458	−	0.970907i	$-0.576970\pi$
$401$	−9.59029	−0.478916	−0.239458	+	0.970907i	$0.576970\pi$
$402$	0	0
$403$	−1.17777	−0.0586691
$404$	0	0
$405$	12.6253	0.627358
$406$	0	0
$407$	−28.6177	−1.41853
$408$	0	0
$409$	19.5036	0.964390	0.482195	−	0.876064i	$-0.339840\pi$
$409$	19.5036	0.964390	0.482195	+	0.876064i	$0.339840\pi$
$410$	0	0
$411$	−13.9228	−0.686763
$412$	0	0
$413$	−9.49852	−0.467391
$414$	0	0
$415$	2.20563	0.108270
$416$	0	0
$417$	−29.1871	−1.42930
$418$	0	0
$419$	−32.7536	−1.60012	−0.800060	−	0.599921i	$-0.795199\pi$
$419$	−32.7536	−1.60012	−0.800060	+	0.599921i	$0.795199\pi$
$420$	0	0
$421$	−7.30658	−0.356101	−0.178051	−	0.984021i	$-0.556979\pi$
$421$	−7.30658	−0.356101	−0.178051	+	0.984021i	$0.556979\pi$
$422$	0	0
$423$	18.4322	0.896203
$424$	0	0
$425$	−17.9217	−0.869332
$426$	0	0
$427$	−38.4501	−1.86073
$428$	0	0
$429$	−3.98643	−0.192467
$430$	0	0
$431$	13.3346	0.642307	0.321153	−	0.947027i	$-0.395929\pi$
$431$	13.3346	0.642307	0.321153	+	0.947027i	$0.395929\pi$
$432$	0	0
$433$	−16.0887	−0.773175	−0.386587	−	0.922253i	$-0.626346\pi$
$433$	−16.0887	−0.773175	−0.386587	+	0.922253i	$0.626346\pi$
$434$	0	0
$435$	−15.9496	−0.764725
$436$	0	0
$437$	−5.32558	−0.254757
$438$	0	0
$439$	3.84338	0.183434	0.0917172	−	0.995785i	$-0.470764\pi$
$439$	3.84338	0.183434	0.0917172	+	0.995785i	$0.470764\pi$
$440$	0	0
$441$	14.2414	0.678162
$442$	0	0
$443$	−14.4500	−0.686542	−0.343271	−	0.939236i	$-0.611535\pi$
$443$	−14.4500	−0.686542	−0.343271	+	0.939236i	$0.611535\pi$
$444$	0	0
$445$	−16.5639	−0.785204
$446$	0	0
$447$	−33.6175	−1.59005
$448$	0	0
$449$	−9.12580	−0.430673	−0.215337	−	0.976540i	$-0.569085\pi$
$449$	−9.12580	−0.430673	−0.215337	+	0.976540i	$0.569085\pi$
$450$	0	0
$451$	−25.8843	−1.21884
$452$	0	0
$453$	19.0601	0.895523
$454$	0	0
$455$	1.51612	0.0710766
$456$	0	0
$457$	0.218873	0.0102384	0.00511922	−	0.999987i	$-0.498370\pi$
$457$	0.218873	0.0102384	0.00511922	+	0.999987i	$0.498370\pi$
$458$	0	0
$459$	−14.8450	−0.692907
$460$	0	0
$461$	−13.1822	−0.613955	−0.306978	−	0.951717i	$-0.599318\pi$
$461$	−13.1822	−0.613955	−0.306978	+	0.951717i	$0.599318\pi$
$462$	0	0
$463$	−7.77002	−0.361104	−0.180552	−	0.983565i	$-0.557788\pi$
$463$	−7.77002	−0.361104	−0.180552	+	0.983565i	$0.557788\pi$
$464$	0	0
$465$	8.40184	0.389626
$466$	0	0
$467$	−16.1107	−0.745515	−0.372758	−	0.927929i	$-0.621588\pi$
$467$	−16.1107	−0.745515	−0.372758	+	0.927929i	$0.621588\pi$
$468$	0	0
$469$	11.2197	0.518078
$470$	0	0
$471$	1.68563	0.0776697
$472$	0	0
$473$	54.6768	2.51404
$474$	0	0
$475$	−6.47669	−0.297171
$476$	0	0
$477$	17.6920	0.810063
$478$	0	0
$479$	−0.985380	−0.0450231	−0.0225116	−	0.999747i	$-0.507166\pi$
$479$	−0.985380	−0.0450231	−0.0225116	+	0.999747i	$0.507166\pi$
$480$	0	0
$481$	1.72343	0.0785819
$482$	0	0
$483$	26.3987	1.20118
$484$	0	0
$485$	−6.70285	−0.304361
$486$	0	0
$487$	12.8005	0.580045	0.290023	−	0.957020i	$-0.406337\pi$
$487$	12.8005	0.580045	0.290023	+	0.957020i	$0.406337\pi$
$488$	0	0
$489$	50.7561	2.29527
$490$	0	0
$491$	−11.4113	−0.514985	−0.257492	−	0.966280i	$-0.582896\pi$
$491$	−11.4113	−0.514985	−0.257492	+	0.966280i	$0.582896\pi$
$492$	0	0
$493$	31.9304	1.43807
$494$	0	0
$495$	9.67606	0.434907
$496$	0	0
$497$	−28.6742	−1.28621
$498$	0	0
$499$	1.85189	0.0829020	0.0414510	−	0.999141i	$-0.486802\pi$
$499$	1.85189	0.0829020	0.0414510	+	0.999141i	$0.486802\pi$
$500$	0	0
$501$	−27.5355	−1.23019
$502$	0	0
$503$	18.2301	0.812839	0.406419	−	0.913687i	$-0.366777\pi$
$503$	18.2301	0.812839	0.406419	+	0.913687i	$0.366777\pi$
$504$	0	0
$505$	3.81946	0.169964
$506$	0	0
$507$	−27.4813	−1.22049
$508$	0	0
$509$	30.9539	1.37201	0.686003	−	0.727598i	$-0.259364\pi$
$509$	30.9539	1.37201	0.686003	+	0.727598i	$0.259364\pi$
$510$	0	0
$511$	−26.7862	−1.18495
$512$	0	0
$513$	−5.36481	−0.236862
$514$	0	0
$515$	−1.85998	−0.0819605
$516$	0	0
$517$	−66.3753	−2.91918
$518$	0	0
$519$	27.8732	1.22350
$520$	0	0
$521$	29.0190	1.27134	0.635672	−	0.771959i	$-0.280723\pi$
$521$	29.0190	1.27134	0.635672	+	0.771959i	$0.280723\pi$
$522$	0	0
$523$	10.2059	0.446273	0.223137	−	0.974787i	$-0.428370\pi$
$523$	10.2059	0.446273	0.223137	+	0.974787i	$0.428370\pi$
$524$	0	0
$525$	32.1047	1.40117
$526$	0	0
$527$	−16.8201	−0.732695
$528$	0	0
$529$	−13.5424	−0.588799
$530$	0	0
$531$	3.65075	0.158429
$532$	0	0
$533$	1.55882	0.0675200
$534$	0	0
$535$	4.08405	0.176569
$536$	0	0
$537$	19.6251	0.846884
$538$	0	0
$539$	−51.2841	−2.20896
$540$	0	0
$541$	39.6458	1.70451	0.852253	−	0.523129i	$-0.175235\pi$
$541$	39.6458	1.70451	0.852253	+	0.523129i	$0.175235\pi$
$542$	0	0
$543$	1.30877	0.0561647
$544$	0	0
$545$	0.601158	0.0257508
$546$	0	0
$547$	−18.0926	−0.773584	−0.386792	−	0.922167i	$-0.626417\pi$
$547$	−18.0926	−0.773584	−0.386792	+	0.922167i	$0.626417\pi$
$548$	0	0
$549$	14.7783	0.630722
$550$	0	0
$551$	11.5392	0.491588
$552$	0	0
$553$	34.7255	1.47668
$554$	0	0
$555$	−12.2944	−0.521868
$556$	0	0
$557$	−25.5133	−1.08103	−0.540516	−	0.841334i	$-0.681771\pi$
$557$	−25.5133	−1.08103	−0.540516	+	0.841334i	$0.681771\pi$
$558$	0	0
$559$	−3.29278	−0.139270
$560$	0	0
$561$	−56.9313	−2.40364
$562$	0	0
$563$	−12.7379	−0.536837	−0.268419	−	0.963302i	$-0.586501\pi$
$563$	−12.7379	−0.536837	−0.268419	+	0.963302i	$0.586501\pi$
$564$	0	0
$565$	18.9902	0.798926
$566$	0	0
$567$	45.2779	1.90149
$568$	0	0
$569$	32.3406	1.35579	0.677894	−	0.735160i	$-0.262893\pi$
$569$	32.3406	1.35579	0.677894	+	0.735160i	$0.262893\pi$
$570$	0	0
$571$	31.3992	1.31401	0.657007	−	0.753884i	$-0.271822\pi$
$571$	31.3992	1.31401	0.657007	+	0.753884i	$0.271822\pi$
$572$	0	0
$573$	13.4651	0.562514
$574$	0	0
$575$	11.5019	0.479661
$576$	0	0
$577$	25.3641	1.05592	0.527960	−	0.849269i	$-0.322957\pi$
$577$	25.3641	1.05592	0.527960	+	0.849269i	$0.322957\pi$
$578$	0	0
$579$	23.0494	0.957900
$580$	0	0
$581$	7.91000	0.328162
$582$	0	0
$583$	−63.7101	−2.63860
$584$	0	0
$585$	−0.582719	−0.0240924
$586$	0	0
$587$	−38.0082	−1.56877	−0.784383	−	0.620277i	$-0.787020\pi$
$587$	−38.0082	−1.56877	−0.784383	+	0.620277i	$0.787020\pi$
$588$	0	0
$589$	−6.07858	−0.250463
$590$	0	0
$591$	44.5771	1.83366
$592$	0	0
$593$	28.7010	1.17861	0.589304	−	0.807911i	$-0.299402\pi$
$593$	28.7010	1.17861	0.589304	+	0.807911i	$0.299402\pi$
$594$	0	0
$595$	21.6521	0.887648
$596$	0	0
$597$	30.6312	1.25365
$598$	0	0
$599$	−25.5053	−1.04212	−0.521059	−	0.853520i	$-0.674463\pi$
$599$	−25.5053	−1.04212	−0.521059	+	0.853520i	$0.674463\pi$
$600$	0	0
$601$	3.29859	0.134552	0.0672762	−	0.997734i	$-0.478569\pi$
$601$	3.29859	0.134552	0.0672762	+	0.997734i	$0.478569\pi$
$602$	0	0
$603$	−4.31229	−0.175610
$604$	0	0
$605$	−22.4968	−0.914627
$606$	0	0
$607$	−6.84359	−0.277773	−0.138886	−	0.990308i	$-0.544352\pi$
$607$	−6.84359	−0.277773	−0.138886	+	0.990308i	$0.544352\pi$
$608$	0	0
$609$	−57.1996	−2.31785
$610$	0	0
$611$	3.99730	0.161714
$612$	0	0
$613$	−33.1661	−1.33957	−0.669784	−	0.742556i	$-0.733613\pi$
$613$	−33.1661	−1.33957	−0.669784	+	0.742556i	$0.733613\pi$
$614$	0	0
$615$	−11.1201	−0.448405
$616$	0	0
$617$	21.2071	0.853766	0.426883	−	0.904307i	$-0.359612\pi$
$617$	21.2071	0.853766	0.426883	+	0.904307i	$0.359612\pi$
$618$	0	0
$619$	34.1470	1.37249	0.686243	−	0.727373i	$-0.259259\pi$
$619$	34.1470	1.37249	0.686243	+	0.727373i	$0.259259\pi$
$620$	0	0
$621$	9.52729	0.382317
$622$	0	0
$623$	−59.4026	−2.37992
$624$	0	0
$625$	7.68823	0.307529
$626$	0	0
$627$	−20.5742	−0.821656
$628$	0	0
$629$	24.6128	0.981377
$630$	0	0
$631$	−1.70207	−0.0677584	−0.0338792	−	0.999426i	$-0.510786\pi$
$631$	−1.70207	−0.0677584	−0.0338792	+	0.999426i	$0.510786\pi$
$632$	0	0
$633$	36.1556	1.43705
$634$	0	0
$635$	−10.1678	−0.403498
$636$	0	0
$637$	3.08847	0.122370
$638$	0	0
$639$	11.0209	0.435980
$640$	0	0
$641$	12.0794	0.477107	0.238553	−	0.971129i	$-0.423327\pi$
$641$	12.0794	0.477107	0.238553	+	0.971129i	$0.423327\pi$
$642$	0	0
$643$	−43.9208	−1.73207	−0.866033	−	0.499987i	$-0.833338\pi$
$643$	−43.9208	−1.73207	−0.866033	+	0.499987i	$0.833338\pi$
$644$	0	0
$645$	23.4896	0.924902
$646$	0	0
$647$	−30.1591	−1.18567	−0.592837	−	0.805322i	$-0.701992\pi$
$647$	−30.1591	−1.18567	−0.592837	+	0.805322i	$0.701992\pi$
$648$	0	0
$649$	−13.1466	−0.516048
$650$	0	0
$651$	30.1313	1.18094
$652$	0	0
$653$	−50.5228	−1.97711	−0.988555	−	0.150862i	$-0.951795\pi$
$653$	−50.5228	−1.97711	−0.988555	+	0.150862i	$0.951795\pi$
$654$	0	0
$655$	−12.0101	−0.469274
$656$	0	0
$657$	10.2953	0.401656
$658$	0	0
$659$	43.4812	1.69379	0.846893	−	0.531763i	$-0.178470\pi$
$659$	43.4812	1.69379	0.846893	+	0.531763i	$0.178470\pi$
$660$	0	0
$661$	14.3401	0.557765	0.278883	−	0.960325i	$-0.410036\pi$
$661$	14.3401	0.557765	0.278883	+	0.960325i	$0.410036\pi$
$662$	0	0
$663$	3.42855	0.133154
$664$	0	0
$665$	7.82478	0.303432
$666$	0	0
$667$	−20.4924	−0.793469
$668$	0	0
$669$	48.8331	1.88800
$670$	0	0
$671$	−53.2175	−2.05444
$672$	0	0
$673$	−22.0505	−0.849982	−0.424991	−	0.905197i	$-0.639723\pi$
$673$	−22.0505	−0.849982	−0.424991	+	0.905197i	$0.639723\pi$
$674$	0	0
$675$	11.5866	0.445968
$676$	0	0
$677$	−27.6304	−1.06192	−0.530960	−	0.847397i	$-0.678169\pi$
$677$	−27.6304	−1.06192	−0.530960	+	0.847397i	$0.678169\pi$
$678$	0	0
$679$	−24.0382	−0.922503
$680$	0	0
$681$	−40.9263	−1.56830
$682$	0	0
$683$	−6.39771	−0.244802	−0.122401	−	0.992481i	$-0.539059\pi$
$683$	−6.39771	−0.244802	−0.122401	+	0.992481i	$0.539059\pi$
$684$	0	0
$685$	7.32879	0.280019
$686$	0	0
$687$	−8.88240	−0.338885
$688$	0	0
$689$	3.83679	0.146170
$690$	0	0
$691$	37.0289	1.40865	0.704324	−	0.709879i	$-0.251250\pi$
$691$	37.0289	1.40865	0.704324	+	0.709879i	$0.251250\pi$
$692$	0	0
$693$	34.7010	1.31818
$694$	0	0
$695$	15.3637	0.582778
$696$	0	0
$697$	22.2619	0.843230
$698$	0	0
$699$	−51.9859	−1.96629
$700$	0	0
$701$	−23.8442	−0.900584	−0.450292	−	0.892881i	$-0.648680\pi$
$701$	−23.8442	−0.900584	−0.450292	+	0.892881i	$0.648680\pi$
$702$	0	0
$703$	8.89476	0.335472
$704$	0	0
$705$	−28.5154	−1.07395
$706$	0	0
$707$	13.6976	0.515153
$708$	0	0
$709$	3.10748	0.116704	0.0583519	−	0.998296i	$-0.481415\pi$
$709$	3.10748	0.116704	0.0583519	+	0.998296i	$0.481415\pi$
$710$	0	0
$711$	−13.3467	−0.500542
$712$	0	0
$713$	10.7949	0.404271
$714$	0	0
$715$	2.09840	0.0784758
$716$	0	0
$717$	18.1015	0.676012
$718$	0	0
$719$	15.6524	0.583736	0.291868	−	0.956459i	$-0.405723\pi$
$719$	15.6524	0.583736	0.291868	+	0.956459i	$0.405723\pi$
$720$	0	0
$721$	−6.67039	−0.248418
$722$	0	0
$723$	−30.7198	−1.14248
$724$	0	0
$725$	−24.9218	−0.925571
$726$	0	0
$727$	12.9463	0.480152	0.240076	−	0.970754i	$-0.422828\pi$
$727$	12.9463	0.480152	0.240076	+	0.970754i	$0.422828\pi$
$728$	0	0
$729$	2.41600	0.0894813
$730$	0	0
$731$	−47.0251	−1.73929
$732$	0	0
$733$	−29.5007	−1.08963	−0.544817	−	0.838555i	$-0.683401\pi$
$733$	−29.5007	−1.08963	−0.544817	+	0.838555i	$0.683401\pi$
$734$	0	0
$735$	−22.0321	−0.812665
$736$	0	0
$737$	15.5288	0.572010
$738$	0	0
$739$	−41.9176	−1.54196	−0.770982	−	0.636857i	$-0.780234\pi$
$739$	−41.9176	−1.54196	−0.770982	+	0.636857i	$0.780234\pi$
$740$	0	0
$741$	1.23904	0.0455171
$742$	0	0
$743$	20.1143	0.737923	0.368961	−	0.929445i	$-0.379713\pi$
$743$	20.1143	0.737923	0.368961	+	0.929445i	$0.379713\pi$
$744$	0	0
$745$	17.6958	0.648323
$746$	0	0
$747$	−3.04020	−0.111235
$748$	0	0
$749$	14.6465	0.535172
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−26.2878	−0.957980
$754$	0	0
$755$	−10.0330	−0.365138
$756$	0	0
$757$	26.2904	0.955539	0.477770	−	0.878485i	$-0.341445\pi$
$757$	26.2904	0.955539	0.477770	+	0.878485i	$0.341445\pi$
$758$	0	0
$759$	36.5375	1.32623
$760$	0	0
$761$	−39.0399	−1.41520	−0.707598	−	0.706615i	$-0.750221\pi$
$761$	−39.0399	−1.41520	−0.707598	+	0.706615i	$0.750221\pi$
$762$	0	0
$763$	2.15591	0.0780493
$764$	0	0
$765$	−8.32195	−0.300881
$766$	0	0
$767$	0.791721	0.0285874
$768$	0	0
$769$	27.5522	0.993559	0.496779	−	0.867877i	$-0.334516\pi$
$769$	27.5522	0.993559	0.496779	+	0.867877i	$0.334516\pi$
$770$	0	0
$771$	−45.1496	−1.62602
$772$	0	0
$773$	7.14155	0.256864	0.128432	−	0.991718i	$-0.459006\pi$
$773$	7.14155	0.256864	0.128432	+	0.991718i	$0.459006\pi$
$774$	0	0
$775$	13.1281	0.471577
$776$	0	0
$777$	−44.0910	−1.58176
$778$	0	0
$779$	8.04517	0.288248
$780$	0	0
$781$	−39.6869	−1.42011
$782$	0	0
$783$	−20.6433	−0.737732
$784$	0	0
$785$	−0.887293	−0.0316688
$786$	0	0
$787$	23.1663	0.825788	0.412894	−	0.910779i	$-0.364518\pi$
$787$	23.1663	0.825788	0.412894	+	0.910779i	$0.364518\pi$
$788$	0	0
$789$	36.2814	1.29165
$790$	0	0
$791$	68.1042	2.42151
$792$	0	0
$793$	3.20490	0.113809
$794$	0	0
$795$	−27.3704	−0.970727
$796$	0	0
$797$	1.70129	0.0602629	0.0301315	−	0.999546i	$-0.490407\pi$
$797$	1.70129	0.0602629	0.0301315	+	0.999546i	$0.490407\pi$
$798$	0	0
$799$	57.0865	2.01957
$800$	0	0
$801$	22.8314	0.806707
$802$	0	0
$803$	−37.0738	−1.30831
$804$	0	0
$805$	−13.8959	−0.489767
$806$	0	0
$807$	−29.8447	−1.05058
$808$	0	0
$809$	38.5226	1.35438	0.677192	−	0.735807i	$-0.263197\pi$
$809$	38.5226	1.35438	0.677192	+	0.735807i	$0.263197\pi$
$810$	0	0
$811$	−8.20547	−0.288133	−0.144067	−	0.989568i	$-0.546018\pi$
$811$	−8.20547	−0.288133	−0.144067	+	0.989568i	$0.546018\pi$
$812$	0	0
$813$	−23.8479	−0.836383
$814$	0	0
$815$	−26.7173	−0.935867
$816$	0	0
$817$	−16.9943	−0.594555
$818$	0	0
$819$	−2.08979	−0.0730231
$820$	0	0
$821$	15.1611	0.529125	0.264562	−	0.964369i	$-0.414772\pi$
$821$	15.1611	0.529125	0.264562	+	0.964369i	$0.414772\pi$
$822$	0	0
$823$	−28.7968	−1.00379	−0.501896	−	0.864928i	$-0.667364\pi$
$823$	−28.7968	−1.00379	−0.501896	+	0.864928i	$0.667364\pi$
$824$	0	0
$825$	44.4350	1.54703
$826$	0	0
$827$	−35.8128	−1.24533	−0.622666	−	0.782488i	$-0.713950\pi$
$827$	−35.8128	−1.24533	−0.622666	+	0.782488i	$0.713950\pi$
$828$	0	0
$829$	−51.0514	−1.77309	−0.886544	−	0.462645i	$-0.846900\pi$
$829$	−51.0514	−1.77309	−0.886544	+	0.462645i	$0.846900\pi$
$830$	0	0
$831$	26.3492	0.914045
$832$	0	0
$833$	44.1072	1.52822
$834$	0	0
$835$	14.4943	0.501596
$836$	0	0
$837$	10.8744	0.375873
$838$	0	0
$839$	27.5439	0.950920	0.475460	−	0.879737i	$-0.342282\pi$
$839$	27.5439	0.950920	0.475460	+	0.879737i	$0.342282\pi$
$840$	0	0
$841$	15.4021	0.531105
$842$	0	0
$843$	16.6874	0.574746
$844$	0	0
$845$	14.4658	0.497639
$846$	0	0
$847$	−80.6798	−2.77219
$848$	0	0
$849$	16.8866	0.579548
$850$	0	0
$851$	−15.7961	−0.541483
$852$	0	0
$853$	52.7334	1.80556	0.902778	−	0.430107i	$-0.141524\pi$
$853$	52.7334	1.80556	0.902778	+	0.430107i	$0.141524\pi$
$854$	0	0
$855$	−3.00745	−0.102853
$856$	0	0
$857$	20.2839	0.692885	0.346443	−	0.938071i	$-0.387390\pi$
$857$	20.2839	0.692885	0.346443	+	0.938071i	$0.387390\pi$
$858$	0	0
$859$	8.72502	0.297694	0.148847	−	0.988860i	$-0.452444\pi$
$859$	8.72502	0.297694	0.148847	+	0.988860i	$0.452444\pi$
$860$	0	0
$861$	−39.8796	−1.35909
$862$	0	0
$863$	−20.7641	−0.706817	−0.353408	−	0.935469i	$-0.614977\pi$
$863$	−20.7641	−0.706817	−0.353408	+	0.935469i	$0.614977\pi$
$864$	0	0
$865$	−14.6721	−0.498865
$866$	0	0
$867$	12.7130	0.431756
$868$	0	0
$869$	48.0624	1.63041
$870$	0	0
$871$	−0.935186	−0.0316876
$872$	0	0
$873$	9.23909	0.312696
$874$	0	0
$875$	−39.4921	−1.33508
$876$	0	0
$877$	45.9513	1.55167	0.775833	−	0.630939i	$-0.217330\pi$
$877$	45.9513	1.55167	0.775833	+	0.630939i	$0.217330\pi$
$878$	0	0
$879$	11.8675	0.400282
$880$	0	0
$881$	−44.3695	−1.49485	−0.747423	−	0.664349i	$-0.768709\pi$
$881$	−44.3695	−1.49485	−0.747423	+	0.664349i	$0.768709\pi$
$882$	0	0
$883$	34.4308	1.15869	0.579345	−	0.815083i	$-0.303309\pi$
$883$	34.4308	1.15869	0.579345	+	0.815083i	$0.303309\pi$
$884$	0	0
$885$	−5.64787	−0.189851
$886$	0	0
$887$	9.46631	0.317848	0.158924	−	0.987291i	$-0.449198\pi$
$887$	9.46631	0.317848	0.158924	+	0.987291i	$0.449198\pi$
$888$	0	0
$889$	−36.4646	−1.22298
$890$	0	0
$891$	62.6675	2.09944
$892$	0	0
$893$	20.6303	0.690368
$894$	0	0
$895$	−10.3304	−0.345306
$896$	0	0
$897$	−2.20039	−0.0734688
$898$	0	0
$899$	−23.3898	−0.780095
$900$	0	0
$901$	54.7942	1.82546
$902$	0	0
$903$	84.2401	2.80333
$904$	0	0
$905$	−0.688919	−0.0229004
$906$	0	0
$907$	49.1029	1.63043	0.815217	−	0.579156i	$-0.196618\pi$
$907$	49.1029	1.63043	0.815217	+	0.579156i	$0.196618\pi$
$908$	0	0
$909$	−5.26468	−0.174618
$910$	0	0
$911$	17.8459	0.591260	0.295630	−	0.955302i	$-0.404470\pi$
$911$	17.8459	0.591260	0.295630	+	0.955302i	$0.404470\pi$
$912$	0	0
$913$	10.9479	0.362324
$914$	0	0
$915$	−22.8627	−0.755816
$916$	0	0
$917$	−43.0715	−1.42235
$918$	0	0
$919$	−51.0203	−1.68300	−0.841501	−	0.540255i	$-0.818328\pi$
$919$	−51.0203	−1.68300	−0.841501	+	0.540255i	$0.818328\pi$
$920$	0	0
$921$	3.29258	0.108494
$922$	0	0
$923$	2.39005	0.0786695
$924$	0	0
$925$	−19.2104	−0.631633
$926$	0	0
$927$	2.56376	0.0842050
$928$	0	0
$929$	31.6458	1.03827	0.519133	−	0.854693i	$-0.326255\pi$
$929$	31.6458	1.03827	0.519133	+	0.854693i	$0.326255\pi$
$930$	0	0
$931$	15.9398	0.522406
$932$	0	0
$933$	−54.7379	−1.79204
$934$	0	0
$935$	29.9678	0.980053
$936$	0	0
$937$	3.25942	0.106481	0.0532404	−	0.998582i	$-0.483045\pi$
$937$	3.25942	0.106481	0.0532404	+	0.998582i	$0.483045\pi$
$938$	0	0
$939$	−40.7139	−1.32865
$940$	0	0
$941$	17.3811	0.566608	0.283304	−	0.959030i	$-0.408570\pi$
$941$	17.3811	0.566608	0.283304	+	0.959030i	$0.408570\pi$
$942$	0	0
$943$	−14.2873	−0.465259
$944$	0	0
$945$	−13.9983	−0.455364
$946$	0	0
$947$	45.8162	1.48883	0.744413	−	0.667719i	$-0.232729\pi$
$947$	45.8162	1.48883	0.744413	+	0.667719i	$0.232729\pi$
$948$	0	0
$949$	2.23268	0.0724760
$950$	0	0
$951$	−42.5121	−1.37855
$952$	0	0
$953$	36.8662	1.19421	0.597107	−	0.802162i	$-0.296317\pi$
$953$	36.8662	1.19421	0.597107	+	0.802162i	$0.296317\pi$
$954$	0	0
$955$	−7.08787	−0.229358
$956$	0	0
$957$	−79.1680	−2.55914
$958$	0	0
$959$	26.2830	0.848723
$960$	0	0
$961$	−18.6788	−0.602543
$962$	0	0
$963$	−5.62938	−0.181404
$964$	0	0
$965$	−12.1329	−0.390571
$966$	0	0
$967$	10.7123	0.344485	0.172243	−	0.985055i	$-0.444899\pi$
$967$	10.7123	0.344485	0.172243	+	0.985055i	$0.444899\pi$
$968$	0	0
$969$	17.6950	0.568445
$970$	0	0
$971$	32.4423	1.04112	0.520561	−	0.853824i	$-0.325723\pi$
$971$	32.4423	1.04112	0.520561	+	0.853824i	$0.325723\pi$
$972$	0	0
$973$	55.0984	1.76637
$974$	0	0
$975$	−2.67600	−0.0857005
$976$	0	0
$977$	−17.8203	−0.570121	−0.285060	−	0.958510i	$-0.592014\pi$
$977$	−17.8203	−0.570121	−0.285060	+	0.958510i	$0.592014\pi$
$978$	0	0
$979$	−82.2171	−2.62767
$980$	0	0
$981$	−0.828625	−0.0264559
$982$	0	0
$983$	−2.06406	−0.0658333	−0.0329166	−	0.999458i	$-0.510480\pi$
$983$	−2.06406	−0.0658333	−0.0329166	+	0.999458i	$0.510480\pi$
$984$	0	0
$985$	−23.4648	−0.747649
$986$	0	0
$987$	−102.264	−3.25510
$988$	0	0
$989$	30.1799	0.959666
$990$	0	0
$991$	31.7175	1.00754	0.503770	−	0.863838i	$-0.331946\pi$
$991$	31.7175	1.00754	0.503770	+	0.863838i	$0.331946\pi$
$992$	0	0
$993$	68.6915	2.17986
$994$	0	0
$995$	−16.1238	−0.511160
$996$	0	0
$997$	−53.5470	−1.69585	−0.847925	−	0.530116i	$-0.822148\pi$
$997$	−53.5470	−1.69585	−0.847925	+	0.530116i	$0.822148\pi$
$998$	0	0
$999$	−15.9124	−0.503447

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.39	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.39	✓	50	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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q+2.13242 q^{3} -1.12247 q^{5} -4.02550 q^{7} +1.54720 q^{9} +O(q^{10})\)	\(q+2.13242 q^{3} -1.12247 q^{5} -4.02550 q^{7} +1.54720 q^{9} -5.57155 q^{11} +0.335534 q^{13} -2.39358 q^{15} +4.79184 q^{17} +1.73171 q^{19} -8.58404 q^{21} -3.07533 q^{23} -3.74005 q^{25} -3.09798 q^{27} +6.66349 q^{29} -3.51015 q^{31} -11.8809 q^{33} +4.51852 q^{35} +5.13640 q^{37} +0.715498 q^{39} +4.64579 q^{41} -9.81357 q^{43} -1.73669 q^{45} +11.9133 q^{47} +9.20464 q^{49} +10.2182 q^{51} +11.4349 q^{53} +6.25393 q^{55} +3.69273 q^{57} +2.35959 q^{59} +9.55165 q^{61} -6.22824 q^{63} -0.376628 q^{65} -2.78716 q^{67} -6.55787 q^{69} +7.12313 q^{71} +6.65412 q^{73} -7.97534 q^{75} +22.4283 q^{77} -8.62639 q^{79} -11.2478 q^{81} -1.96497 q^{83} -5.37873 q^{85} +14.2093 q^{87} +14.7566 q^{89} -1.35069 q^{91} -7.48510 q^{93} -1.94380 q^{95} +5.97150 q^{97} -8.62029 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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