LMFDB - Embedded newform 6008.2.a.e.1.17

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			1.17
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)

$f(q)$	$=$	$q-1.03653 q^{3} +2.53205 q^{5} -0.333645 q^{7} -1.92561 q^{9} +O(q^{10})$	$q-1.03653 q^{3} +2.53205 q^{5} -0.333645 q^{7} -1.92561 q^{9} -4.00849 q^{11} -1.11706 q^{13} -2.62454 q^{15} +7.29433 q^{17} -5.63287 q^{19} +0.345833 q^{21} -4.25451 q^{23} +1.41125 q^{25} +5.10553 q^{27} -1.89980 q^{29} +9.43795 q^{31} +4.15491 q^{33} -0.844805 q^{35} -0.479113 q^{37} +1.15787 q^{39} -2.92963 q^{41} -2.67273 q^{43} -4.87573 q^{45} -5.38038 q^{47} -6.88868 q^{49} -7.56078 q^{51} +11.0743 q^{53} -10.1497 q^{55} +5.83863 q^{57} +13.7733 q^{59} +9.57938 q^{61} +0.642470 q^{63} -2.82845 q^{65} -9.91497 q^{67} +4.40992 q^{69} -12.2338 q^{71} +11.3112 q^{73} -1.46280 q^{75} +1.33741 q^{77} +8.09784 q^{79} +0.484799 q^{81} +4.23590 q^{83} +18.4696 q^{85} +1.96919 q^{87} +0.712086 q^{89} +0.372702 q^{91} -9.78270 q^{93} -14.2627 q^{95} +5.75930 q^{97} +7.71878 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.03653	−0.598440	−0.299220	−	0.954184i	$-0.596726\pi$
$3$	−1.03653	−0.598440	−0.299220	+	0.954184i	$0.596726\pi$
$4$	0	0
$5$	2.53205	1.13237	0.566183	−	0.824280i	$-0.308420\pi$
$5$	2.53205	1.13237	0.566183	+	0.824280i	$0.308420\pi$
$6$	0	0
$7$	−0.333645	−0.126106	−0.0630530	−	0.998010i	$-0.520084\pi$
$7$	−0.333645	−0.126106	−0.0630530	+	0.998010i	$0.520084\pi$
$8$	0	0
$9$	−1.92561	−0.641870
$10$	0	0
$11$	−4.00849	−1.20860	−0.604302	−	0.796755i	$-0.706548\pi$
$11$	−4.00849	−1.20860	−0.604302	+	0.796755i	$0.706548\pi$
$12$	0	0
$13$	−1.11706	−0.309817	−0.154909	−	0.987929i	$-0.549508\pi$
$13$	−1.11706	−0.309817	−0.154909	+	0.987929i	$0.549508\pi$
$14$	0	0
$15$	−2.62454	−0.677652
$16$	0	0
$17$	7.29433	1.76914	0.884568	−	0.466412i	$-0.154453\pi$
$17$	7.29433	1.76914	0.884568	+	0.466412i	$0.154453\pi$
$18$	0	0
$19$	−5.63287	−1.29227	−0.646134	−	0.763224i	$-0.723615\pi$
$19$	−5.63287	−1.29227	−0.646134	+	0.763224i	$0.723615\pi$
$20$	0	0
$21$	0.345833	0.0754669
$22$	0	0
$23$	−4.25451	−0.887126	−0.443563	−	0.896243i	$-0.646286\pi$
$23$	−4.25451	−0.887126	−0.443563	+	0.896243i	$0.646286\pi$
$24$	0	0
$25$	1.41125	0.282251
$26$	0	0
$27$	5.10553	0.982560
$28$	0	0
$29$	−1.89980	−0.352784	−0.176392	−	0.984320i	$-0.556443\pi$
$29$	−1.89980	−0.352784	−0.176392	+	0.984320i	$0.556443\pi$
$30$	0	0
$31$	9.43795	1.69511	0.847553	−	0.530711i	$-0.178075\pi$
$31$	9.43795	1.69511	0.847553	+	0.530711i	$0.178075\pi$
$32$	0	0
$33$	4.15491	0.723277
$34$	0	0
$35$	−0.844805	−0.142798
$36$	0	0
$37$	−0.479113	−0.0787657	−0.0393828	−	0.999224i	$-0.512539\pi$
$37$	−0.479113	−0.0787657	−0.0393828	+	0.999224i	$0.512539\pi$
$38$	0	0
$39$	1.15787	0.185407
$40$	0	0
$41$	−2.92963	−0.457532	−0.228766	−	0.973481i	$-0.573469\pi$
$41$	−2.92963	−0.457532	−0.228766	+	0.973481i	$0.573469\pi$
$42$	0	0
$43$	−2.67273	−0.407588	−0.203794	−	0.979014i	$-0.565327\pi$
$43$	−2.67273	−0.407588	−0.203794	+	0.979014i	$0.565327\pi$
$44$	0	0
$45$	−4.87573	−0.726831
$46$	0	0
$47$	−5.38038	−0.784809	−0.392404	−	0.919793i	$-0.628357\pi$
$47$	−5.38038	−0.784809	−0.392404	+	0.919793i	$0.628357\pi$
$48$	0	0
$49$	−6.88868	−0.984097
$50$	0	0
$51$	−7.56078	−1.05872
$52$	0	0
$53$	11.0743	1.52117	0.760584	−	0.649239i	$-0.224913\pi$
$53$	11.0743	1.52117	0.760584	+	0.649239i	$0.224913\pi$
$54$	0	0
$55$	−10.1497	−1.36858
$56$	0	0
$57$	5.83863	0.773345
$58$	0	0
$59$	13.7733	1.79313	0.896564	−	0.442914i	$-0.146055\pi$
$59$	13.7733	1.79313	0.896564	+	0.442914i	$0.146055\pi$
$60$	0	0
$61$	9.57938	1.22651	0.613257	−	0.789883i	$-0.289859\pi$
$61$	9.57938	1.22651	0.613257	+	0.789883i	$0.289859\pi$
$62$	0	0
$63$	0.642470	0.0809437
$64$	0	0
$65$	−2.82845	−0.350826
$66$	0	0
$67$	−9.91497	−1.21131	−0.605653	−	0.795729i	$-0.707088\pi$
$67$	−9.91497	−1.21131	−0.605653	+	0.795729i	$0.707088\pi$
$68$	0	0
$69$	4.40992	0.530892
$70$	0	0
$71$	−12.2338	−1.45189	−0.725945	−	0.687753i	$-0.758597\pi$
$71$	−12.2338	−1.45189	−0.725945	+	0.687753i	$0.758597\pi$
$72$	0	0
$73$	11.3112	1.32387	0.661935	−	0.749561i	$-0.269735\pi$
$73$	11.3112	1.32387	0.661935	+	0.749561i	$0.269735\pi$
$74$	0	0
$75$	−1.46280	−0.168910
$76$	0	0
$77$	1.33741	0.152412
$78$	0	0
$79$	8.09784	0.911079	0.455539	−	0.890216i	$-0.349446\pi$
$79$	8.09784	0.911079	0.455539	+	0.890216i	$0.349446\pi$
$80$	0	0
$81$	0.484799	0.0538665
$82$	0	0
$83$	4.23590	0.464951	0.232475	−	0.972602i	$-0.425318\pi$
$83$	4.23590	0.464951	0.232475	+	0.972602i	$0.425318\pi$
$84$	0	0
$85$	18.4696	2.00331
$86$	0	0
$87$	1.96919	0.211120
$88$	0	0
$89$	0.712086	0.0754809	0.0377405	−	0.999288i	$-0.487984\pi$
$89$	0.712086	0.0754809	0.0377405	+	0.999288i	$0.487984\pi$
$90$	0	0
$91$	0.372702	0.0390698
$92$	0	0
$93$	−9.78270	−1.01442
$94$	0	0
$95$	−14.2627	−1.46332
$96$	0	0
$97$	5.75930	0.584769	0.292384	−	0.956301i	$-0.405551\pi$
$97$	5.75930	0.584769	0.292384	+	0.956301i	$0.405551\pi$
$98$	0	0
$99$	7.71878	0.775767
$100$	0	0
$101$	15.4644	1.53876	0.769381	−	0.638791i	$-0.220565\pi$
$101$	15.4644	1.53876	0.769381	+	0.638791i	$0.220565\pi$
$102$	0	0
$103$	−8.14373	−0.802426	−0.401213	−	0.915985i	$-0.631411\pi$
$103$	−8.14373	−0.802426	−0.401213	+	0.915985i	$0.631411\pi$
$104$	0	0
$105$	0.875664	0.0854561
$106$	0	0
$107$	20.3670	1.96895	0.984475	−	0.175527i	$-0.0561631\pi$
$107$	20.3670	1.96895	0.984475	+	0.175527i	$0.0561631\pi$
$108$	0	0
$109$	8.19367	0.784811	0.392406	−	0.919792i	$-0.371643\pi$
$109$	8.19367	0.784811	0.392406	+	0.919792i	$0.371643\pi$
$110$	0	0
$111$	0.496614	0.0471365
$112$	0	0
$113$	2.82473	0.265728	0.132864	−	0.991134i	$-0.457583\pi$
$113$	2.82473	0.265728	0.132864	+	0.991134i	$0.457583\pi$
$114$	0	0
$115$	−10.7726	−1.00455
$116$	0	0
$117$	2.15102	0.198862
$118$	0	0
$119$	−2.43372	−0.223099
$120$	0	0
$121$	5.06798	0.460725
$122$	0	0
$123$	3.03665	0.273805
$124$	0	0
$125$	−9.08687	−0.812754
$126$	0	0
$127$	16.6619	1.47851	0.739253	−	0.673428i	$-0.235179\pi$
$127$	16.6619	1.47851	0.739253	+	0.673428i	$0.235179\pi$
$128$	0	0
$129$	2.77036	0.243917
$130$	0	0
$131$	9.30397	0.812892	0.406446	−	0.913675i	$-0.366768\pi$
$131$	9.30397	0.812892	0.406446	+	0.913675i	$0.366768\pi$
$132$	0	0
$133$	1.87938	0.162963
$134$	0	0
$135$	12.9274	1.11262
$136$	0	0
$137$	−1.24260	−0.106163	−0.0530814	−	0.998590i	$-0.516904\pi$
$137$	−1.24260	−0.106163	−0.0530814	+	0.998590i	$0.516904\pi$
$138$	0	0
$139$	2.75676	0.233826	0.116913	−	0.993142i	$-0.462700\pi$
$139$	2.75676	0.233826	0.116913	+	0.993142i	$0.462700\pi$
$140$	0	0
$141$	5.57691	0.469661
$142$	0	0
$143$	4.47773	0.374447
$144$	0	0
$145$	−4.81037	−0.399480
$146$	0	0
$147$	7.14031	0.588923
$148$	0	0
$149$	16.3242	1.33733	0.668663	−	0.743565i	$-0.266867\pi$
$149$	16.3242	1.33733	0.668663	+	0.743565i	$0.266867\pi$
$150$	0	0
$151$	8.82551	0.718210	0.359105	−	0.933297i	$-0.383082\pi$
$151$	8.82551	0.718210	0.359105	+	0.933297i	$0.383082\pi$
$152$	0	0
$153$	−14.0460	−1.13555
$154$	0	0
$155$	23.8973	1.91948
$156$	0	0
$157$	−9.71273	−0.775160	−0.387580	−	0.921836i	$-0.626689\pi$
$157$	−9.71273	−0.775160	−0.387580	+	0.921836i	$0.626689\pi$
$158$	0	0
$159$	−11.4788	−0.910328
$160$	0	0
$161$	1.41950	0.111872
$162$	0	0
$163$	−18.7758	−1.47063	−0.735317	−	0.677723i	$-0.762967\pi$
$163$	−18.7758	−1.47063	−0.735317	+	0.677723i	$0.762967\pi$
$164$	0	0
$165$	10.5204	0.819014
$166$	0	0
$167$	−24.4140	−1.88921	−0.944607	−	0.328202i	$-0.893557\pi$
$167$	−24.4140	−1.88921	−0.944607	+	0.328202i	$0.893557\pi$
$168$	0	0
$169$	−11.7522	−0.904013
$170$	0	0
$171$	10.8467	0.829468
$172$	0	0
$173$	−11.3839	−0.865505	−0.432753	−	0.901513i	$-0.642458\pi$
$173$	−11.3839	−0.865505	−0.432753	+	0.901513i	$0.642458\pi$
$174$	0	0
$175$	−0.470858	−0.0355935
$176$	0	0
$177$	−14.2764	−1.07308
$178$	0	0
$179$	−3.39117	−0.253468	−0.126734	−	0.991937i	$-0.540449\pi$
$179$	−3.39117	−0.253468	−0.126734	+	0.991937i	$0.540449\pi$
$180$	0	0
$181$	−16.5572	−1.23069	−0.615344	−	0.788259i	$-0.710983\pi$
$181$	−16.5572	−1.23069	−0.615344	+	0.788259i	$0.710983\pi$
$182$	0	0
$183$	−9.92930	−0.733995
$184$	0	0
$185$	−1.21314	−0.0891915
$186$	0	0
$187$	−29.2392	−2.13819
$188$	0	0
$189$	−1.70344	−0.123907
$190$	0	0
$191$	10.9196	0.790116	0.395058	−	0.918656i	$-0.370725\pi$
$191$	10.9196	0.790116	0.395058	+	0.918656i	$0.370725\pi$
$192$	0	0
$193$	14.3531	1.03316	0.516580	−	0.856239i	$-0.327205\pi$
$193$	14.3531	1.03316	0.516580	+	0.856239i	$0.327205\pi$
$194$	0	0
$195$	2.93177	0.209948
$196$	0	0
$197$	1.40276	0.0999423	0.0499712	−	0.998751i	$-0.484087\pi$
$197$	1.40276	0.0999423	0.0499712	+	0.998751i	$0.484087\pi$
$198$	0	0
$199$	7.37657	0.522911	0.261455	−	0.965216i	$-0.415798\pi$
$199$	7.37657	0.522911	0.261455	+	0.965216i	$0.415798\pi$
$200$	0	0
$201$	10.2771	0.724894
$202$	0	0
$203$	0.633858	0.0444881
$204$	0	0
$205$	−7.41796	−0.518093
$206$	0	0
$207$	8.19252	0.569420
$208$	0	0
$209$	22.5793	1.56184
$210$	0	0
$211$	13.8794	0.955499	0.477750	−	0.878496i	$-0.341453\pi$
$211$	13.8794	0.955499	0.477750	+	0.878496i	$0.341453\pi$
$212$	0	0
$213$	12.6807	0.868869
$214$	0	0
$215$	−6.76748	−0.461538
$216$	0	0
$217$	−3.14893	−0.213763
$218$	0	0
$219$	−11.7243	−0.792257
$220$	0	0
$221$	−8.14822	−0.548109
$222$	0	0
$223$	−11.2011	−0.750082	−0.375041	−	0.927008i	$-0.622371\pi$
$223$	−11.2011	−0.750082	−0.375041	+	0.927008i	$0.622371\pi$
$224$	0	0
$225$	−2.71752	−0.181168
$226$	0	0
$227$	26.6371	1.76797	0.883985	−	0.467516i	$-0.154851\pi$
$227$	26.6371	1.76797	0.883985	+	0.467516i	$0.154851\pi$
$228$	0	0
$229$	17.0506	1.12673	0.563367	−	0.826207i	$-0.309506\pi$
$229$	17.0506	1.12673	0.563367	+	0.826207i	$0.309506\pi$
$230$	0	0
$231$	−1.38627	−0.0912096
$232$	0	0
$233$	−16.6402	−1.09013	−0.545067	−	0.838393i	$-0.683496\pi$
$233$	−16.6402	−1.09013	−0.545067	+	0.838393i	$0.683496\pi$
$234$	0	0
$235$	−13.6234	−0.888690
$236$	0	0
$237$	−8.39364	−0.545226
$238$	0	0
$239$	4.49038	0.290459	0.145229	−	0.989398i	$-0.453608\pi$
$239$	4.49038	0.290459	0.145229	+	0.989398i	$0.453608\pi$
$240$	0	0
$241$	−5.12770	−0.330304	−0.165152	−	0.986268i	$-0.552812\pi$
$241$	−5.12770	−0.330304	−0.165152	+	0.986268i	$0.552812\pi$
$242$	0	0
$243$	−15.8191	−1.01480
$244$	0	0
$245$	−17.4425	−1.11436
$246$	0	0
$247$	6.29226	0.400367
$248$	0	0
$249$	−4.39063	−0.278245
$250$	0	0
$251$	21.6896	1.36903	0.684517	−	0.728997i	$-0.260013\pi$
$251$	21.6896	1.36903	0.684517	+	0.728997i	$0.260013\pi$
$252$	0	0
$253$	17.0541	1.07219
$254$	0	0
$255$	−19.1442	−1.19886
$256$	0	0
$257$	−12.7772	−0.797019	−0.398510	−	0.917164i	$-0.630472\pi$
$257$	−12.7772	−0.797019	−0.398510	+	0.917164i	$0.630472\pi$
$258$	0	0
$259$	0.159854	0.00993283
$260$	0	0
$261$	3.65827	0.226441
$262$	0	0
$263$	−4.02954	−0.248472	−0.124236	−	0.992253i	$-0.539648\pi$
$263$	−4.02954	−0.248472	−0.124236	+	0.992253i	$0.539648\pi$
$264$	0	0
$265$	28.0406	1.72252
$266$	0	0
$267$	−0.738097	−0.0451708
$268$	0	0
$269$	−21.8554	−1.33254	−0.666272	−	0.745709i	$-0.732111\pi$
$269$	−21.8554	−1.33254	−0.666272	+	0.745709i	$0.732111\pi$
$270$	0	0
$271$	−6.58932	−0.400273	−0.200136	−	0.979768i	$-0.564139\pi$
$271$	−6.58932	−0.400273	−0.200136	+	0.979768i	$0.564139\pi$
$272$	0	0
$273$	−0.386317	−0.0233809
$274$	0	0
$275$	−5.65699	−0.341129
$276$	0	0
$277$	18.7032	1.12377	0.561884	−	0.827216i	$-0.310077\pi$
$277$	18.7032	1.12377	0.561884	+	0.827216i	$0.310077\pi$
$278$	0	0
$279$	−18.1738	−1.08804
$280$	0	0
$281$	22.1897	1.32373	0.661863	−	0.749625i	$-0.269766\pi$
$281$	22.1897	1.32373	0.661863	+	0.749625i	$0.269766\pi$
$282$	0	0
$283$	−11.2094	−0.666328	−0.333164	−	0.942869i	$-0.608116\pi$
$283$	−11.2094	−0.666328	−0.333164	+	0.942869i	$0.608116\pi$
$284$	0	0
$285$	14.7837	0.875709
$286$	0	0
$287$	0.977458	0.0576975
$288$	0	0
$289$	36.2073	2.12984
$290$	0	0
$291$	−5.96968	−0.349949
$292$	0	0
$293$	12.8818	0.752564	0.376282	−	0.926505i	$-0.377202\pi$
$293$	12.8818	0.752564	0.376282	+	0.926505i	$0.377202\pi$
$294$	0	0
$295$	34.8746	2.03048
$296$	0	0
$297$	−20.4655	−1.18753
$298$	0	0
$299$	4.75255	0.274847
$300$	0	0
$301$	0.891745	0.0513993
$302$	0	0
$303$	−16.0292	−0.920856
$304$	0	0
$305$	24.2554	1.38886
$306$	0	0
$307$	19.4035	1.10741	0.553707	−	0.832711i	$-0.313213\pi$
$307$	19.4035	1.10741	0.553707	+	0.832711i	$0.313213\pi$
$308$	0	0
$309$	8.44121	0.480204
$310$	0	0
$311$	20.6297	1.16980	0.584901	−	0.811104i	$-0.301133\pi$
$311$	20.6297	1.16980	0.584901	+	0.811104i	$0.301133\pi$
$312$	0	0
$313$	23.7872	1.34453	0.672266	−	0.740310i	$-0.265321\pi$
$313$	23.7872	1.34453	0.672266	+	0.740310i	$0.265321\pi$
$314$	0	0
$315$	1.62676	0.0916578
$316$	0	0
$317$	27.4129	1.53966	0.769831	−	0.638248i	$-0.220341\pi$
$317$	27.4129	1.53966	0.769831	+	0.638248i	$0.220341\pi$
$318$	0	0
$319$	7.61532	0.426376
$320$	0	0
$321$	−21.1109	−1.17830
$322$	0	0
$323$	−41.0880	−2.28620
$324$	0	0
$325$	−1.57646	−0.0874461
$326$	0	0
$327$	−8.49297	−0.469662
$328$	0	0
$329$	1.79514	0.0989691
$330$	0	0
$331$	−21.3830	−1.17532	−0.587658	−	0.809110i	$-0.699950\pi$
$331$	−21.3830	−1.17532	−0.587658	+	0.809110i	$0.699950\pi$
$332$	0	0
$333$	0.922584	0.0505573
$334$	0	0
$335$	−25.1052	−1.37164
$336$	0	0
$337$	8.03958	0.437944	0.218972	−	0.975731i	$-0.429730\pi$
$337$	8.03958	0.437944	0.218972	+	0.975731i	$0.429730\pi$
$338$	0	0
$339$	−2.92791	−0.159022
$340$	0	0
$341$	−37.8319	−2.04871
$342$	0	0
$343$	4.63389	0.250207
$344$	0	0
$345$	11.1661	0.601163
$346$	0	0
$347$	−3.40048	−0.182547	−0.0912737	−	0.995826i	$-0.529094\pi$
$347$	−3.40048	−0.182547	−0.0912737	+	0.995826i	$0.529094\pi$
$348$	0	0
$349$	8.80924	0.471548	0.235774	−	0.971808i	$-0.424238\pi$
$349$	8.80924	0.471548	0.235774	+	0.971808i	$0.424238\pi$
$350$	0	0
$351$	−5.70320	−0.304414
$352$	0	0
$353$	21.4924	1.14393	0.571963	−	0.820280i	$-0.306182\pi$
$353$	21.4924	1.14393	0.571963	+	0.820280i	$0.306182\pi$
$354$	0	0
$355$	−30.9766	−1.64407
$356$	0	0
$357$	2.52262	0.133511
$358$	0	0
$359$	15.6038	0.823537	0.411769	−	0.911288i	$-0.364911\pi$
$359$	15.6038	0.823537	0.411769	+	0.911288i	$0.364911\pi$
$360$	0	0
$361$	12.7292	0.669958
$362$	0	0
$363$	−5.25310	−0.275716
$364$	0	0
$365$	28.6403	1.49910
$366$	0	0
$367$	−20.1545	−1.05206	−0.526028	−	0.850467i	$-0.676319\pi$
$367$	−20.1545	−1.05206	−0.526028	+	0.850467i	$0.676319\pi$
$368$	0	0
$369$	5.64133	0.293676
$370$	0	0
$371$	−3.69488	−0.191829
$372$	0	0
$373$	−35.8287	−1.85514	−0.927570	−	0.373649i	$-0.878107\pi$
$373$	−35.8287	−1.85514	−0.927570	+	0.373649i	$0.878107\pi$
$374$	0	0
$375$	9.41880	0.486385
$376$	0	0
$377$	2.12219	0.109298
$378$	0	0
$379$	4.98404	0.256013	0.128007	−	0.991773i	$-0.459142\pi$
$379$	4.98404	0.256013	0.128007	+	0.991773i	$0.459142\pi$
$380$	0	0
$381$	−17.2705	−0.884797
$382$	0	0
$383$	−10.5401	−0.538572	−0.269286	−	0.963060i	$-0.586788\pi$
$383$	−10.5401	−0.538572	−0.269286	+	0.963060i	$0.586788\pi$
$384$	0	0
$385$	3.38639	0.172586
$386$	0	0
$387$	5.14664	0.261618
$388$	0	0
$389$	−13.0652	−0.662432	−0.331216	−	0.943555i	$-0.607459\pi$
$389$	−13.0652	−0.662432	−0.331216	+	0.943555i	$0.607459\pi$
$390$	0	0
$391$	−31.0338	−1.56945
$392$	0	0
$393$	−9.64383	−0.486467
$394$	0	0
$395$	20.5041	1.03167
$396$	0	0
$397$	31.1313	1.56244	0.781218	−	0.624259i	$-0.214599\pi$
$397$	31.1313	1.56244	0.781218	+	0.624259i	$0.214599\pi$
$398$	0	0
$399$	−1.94803	−0.0975235
$400$	0	0
$401$	−19.0218	−0.949904	−0.474952	−	0.880012i	$-0.657535\pi$
$401$	−19.0218	−0.949904	−0.474952	+	0.880012i	$0.657535\pi$
$402$	0	0
$403$	−10.5428	−0.525173
$404$	0	0
$405$	1.22753	0.0609966
$406$	0	0
$407$	1.92052	0.0951965
$408$	0	0
$409$	−14.4726	−0.715624	−0.357812	−	0.933794i	$-0.616477\pi$
$409$	−14.4726	−0.715624	−0.357812	+	0.933794i	$0.616477\pi$
$410$	0	0
$411$	1.28799	0.0635320
$412$	0	0
$413$	−4.59539	−0.226124
$414$	0	0
$415$	10.7255	0.526494
$416$	0	0
$417$	−2.85746	−0.139930
$418$	0	0
$419$	22.5646	1.10235	0.551177	−	0.834388i	$-0.314179\pi$
$419$	22.5646	1.10235	0.551177	+	0.834388i	$0.314179\pi$
$420$	0	0
$421$	−23.5020	−1.14542	−0.572709	−	0.819759i	$-0.694108\pi$
$421$	−23.5020	−1.14542	−0.572709	+	0.819759i	$0.694108\pi$
$422$	0	0
$423$	10.3605	0.503745
$424$	0	0
$425$	10.2941	0.499340
$426$	0	0
$427$	−3.19612	−0.154671
$428$	0	0
$429$	−4.64129	−0.224084
$430$	0	0
$431$	11.6189	0.559664	0.279832	−	0.960049i	$-0.409721\pi$
$431$	11.6189	0.559664	0.279832	+	0.960049i	$0.409721\pi$
$432$	0	0
$433$	16.8523	0.809869	0.404934	−	0.914346i	$-0.367294\pi$
$433$	16.8523	0.809869	0.404934	+	0.914346i	$0.367294\pi$
$434$	0	0
$435$	4.98609	0.239065
$436$	0	0
$437$	23.9651	1.14641
$438$	0	0
$439$	−32.6896	−1.56019	−0.780096	−	0.625660i	$-0.784830\pi$
$439$	−32.6896	−1.56019	−0.780096	+	0.625660i	$0.784830\pi$
$440$	0	0
$441$	13.2649	0.631662
$442$	0	0
$443$	14.7830	0.702360	0.351180	−	0.936308i	$-0.385781\pi$
$443$	14.7830	0.702360	0.351180	+	0.936308i	$0.385781\pi$
$444$	0	0
$445$	1.80303	0.0854720
$446$	0	0
$447$	−16.9204	−0.800310
$448$	0	0
$449$	21.6703	1.02269	0.511343	−	0.859377i	$-0.329148\pi$
$449$	21.6703	1.02269	0.511343	+	0.859377i	$0.329148\pi$
$450$	0	0
$451$	11.7434	0.552975
$452$	0	0
$453$	−9.14789	−0.429805
$454$	0	0
$455$	0.943700	0.0442413
$456$	0	0
$457$	28.1372	1.31620	0.658101	−	0.752929i	$-0.271360\pi$
$457$	28.1372	1.31620	0.658101	+	0.752929i	$0.271360\pi$
$458$	0	0
$459$	37.2415	1.73828
$460$	0	0
$461$	−16.4770	−0.767409	−0.383705	−	0.923456i	$-0.625352\pi$
$461$	−16.4770	−0.767409	−0.383705	+	0.923456i	$0.625352\pi$
$462$	0	0
$463$	−36.2859	−1.68635	−0.843174	−	0.537641i	$-0.819315\pi$
$463$	−36.2859	−1.68635	−0.843174	+	0.537641i	$0.819315\pi$
$464$	0	0
$465$	−24.7702	−1.14869
$466$	0	0
$467$	−9.94066	−0.459999	−0.229999	−	0.973191i	$-0.573872\pi$
$467$	−9.94066	−0.459999	−0.229999	+	0.973191i	$0.573872\pi$
$468$	0	0
$469$	3.30808	0.152753
$470$	0	0
$471$	10.0675	0.463887
$472$	0	0
$473$	10.7136	0.492613
$474$	0	0
$475$	−7.94940	−0.364744
$476$	0	0
$477$	−21.3247	−0.976392
$478$	0	0
$479$	−12.1853	−0.556762	−0.278381	−	0.960471i	$-0.589798\pi$
$479$	−12.1853	−0.556762	−0.278381	+	0.960471i	$0.589798\pi$
$480$	0	0
$481$	0.535199	0.0244030
$482$	0	0
$483$	−1.47135	−0.0669487
$484$	0	0
$485$	14.5828	0.662172
$486$	0	0
$487$	23.6953	1.07374	0.536869	−	0.843666i	$-0.319607\pi$
$487$	23.6953	1.07374	0.536869	+	0.843666i	$0.319607\pi$
$488$	0	0
$489$	19.4617	0.880087
$490$	0	0
$491$	27.3741	1.23537	0.617687	−	0.786424i	$-0.288070\pi$
$491$	27.3741	1.23537	0.617687	+	0.786424i	$0.288070\pi$
$492$	0	0
$493$	−13.8578	−0.624122
$494$	0	0
$495$	19.5443	0.878451
$496$	0	0
$497$	4.08176	0.183092
$498$	0	0
$499$	−6.41437	−0.287146	−0.143573	−	0.989640i	$-0.545859\pi$
$499$	−6.41437	−0.287146	−0.143573	+	0.989640i	$0.545859\pi$
$500$	0	0
$501$	25.3058	1.13058
$502$	0	0
$503$	5.49851	0.245167	0.122583	−	0.992458i	$-0.460882\pi$
$503$	5.49851	0.245167	0.122583	+	0.992458i	$0.460882\pi$
$504$	0	0
$505$	39.1565	1.74244
$506$	0	0
$507$	12.1815	0.540998
$508$	0	0
$509$	−18.8142	−0.833924	−0.416962	−	0.908924i	$-0.636905\pi$
$509$	−18.8142	−0.833924	−0.416962	+	0.908924i	$0.636905\pi$
$510$	0	0
$511$	−3.77391	−0.166948
$512$	0	0
$513$	−28.7588	−1.26973
$514$	0	0
$515$	−20.6203	−0.908639
$516$	0	0
$517$	21.5672	0.948523
$518$	0	0
$519$	11.7998	0.517953
$520$	0	0
$521$	−8.75170	−0.383419	−0.191709	−	0.981452i	$-0.561403\pi$
$521$	−8.75170	−0.383419	−0.191709	+	0.981452i	$0.561403\pi$
$522$	0	0
$523$	22.4680	0.982458	0.491229	−	0.871030i	$-0.336548\pi$
$523$	22.4680	0.982458	0.491229	+	0.871030i	$0.336548\pi$
$524$	0	0
$525$	0.488057	0.0213006
$526$	0	0
$527$	68.8435	2.99887
$528$	0	0
$529$	−4.89916	−0.213007
$530$	0	0
$531$	−26.5220	−1.15095
$532$	0	0
$533$	3.27258	0.141751
$534$	0	0
$535$	51.5701	2.22957
$536$	0	0
$537$	3.51505	0.151685
$538$	0	0
$539$	27.6132	1.18938
$540$	0	0
$541$	29.1157	1.25178	0.625891	−	0.779910i	$-0.284735\pi$
$541$	29.1157	1.25178	0.625891	+	0.779910i	$0.284735\pi$
$542$	0	0
$543$	17.1620	0.736493
$544$	0	0
$545$	20.7467	0.888693
$546$	0	0
$547$	45.9869	1.96626	0.983128	−	0.182919i	$-0.0585547\pi$
$547$	45.9869	1.96626	0.983128	+	0.182919i	$0.0585547\pi$
$548$	0	0
$549$	−18.4461	−0.787262
$550$	0	0
$551$	10.7013	0.455891
$552$	0	0
$553$	−2.70181	−0.114893
$554$	0	0
$555$	1.25745	0.0533757
$556$	0	0
$557$	21.3860	0.906153	0.453076	−	0.891472i	$-0.350326\pi$
$557$	21.3860	0.906153	0.453076	+	0.891472i	$0.350326\pi$
$558$	0	0
$559$	2.98561	0.126278
$560$	0	0
$561$	30.3073	1.27958
$562$	0	0
$563$	−36.3102	−1.53029	−0.765146	−	0.643856i	$-0.777333\pi$
$563$	−36.3102	−1.53029	−0.765146	+	0.643856i	$0.777333\pi$
$564$	0	0
$565$	7.15233	0.300901
$566$	0	0
$567$	−0.161751	−0.00679289
$568$	0	0
$569$	32.9668	1.38204	0.691020	−	0.722836i	$-0.257162\pi$
$569$	32.9668	1.38204	0.691020	+	0.722836i	$0.257162\pi$
$570$	0	0
$571$	31.6514	1.32457	0.662284	−	0.749253i	$-0.269587\pi$
$571$	31.6514	1.32457	0.662284	+	0.749253i	$0.269587\pi$
$572$	0	0
$573$	−11.3185	−0.472837
$574$	0	0
$575$	−6.00419	−0.250392
$576$	0	0
$577$	41.4459	1.72542	0.862709	−	0.505701i	$-0.168766\pi$
$577$	41.4459	1.72542	0.862709	+	0.505701i	$0.168766\pi$
$578$	0	0
$579$	−14.8774	−0.618284
$580$	0	0
$581$	−1.41329	−0.0586331
$582$	0	0
$583$	−44.3911	−1.83849
$584$	0	0
$585$	5.44649	0.225185
$586$	0	0
$587$	−25.6584	−1.05903	−0.529517	−	0.848299i	$-0.677627\pi$
$587$	−25.6584	−1.05903	−0.529517	+	0.848299i	$0.677627\pi$
$588$	0	0
$589$	−53.1627	−2.19053
$590$	0	0
$591$	−1.45400	−0.0598095
$592$	0	0
$593$	−18.7863	−0.771460	−0.385730	−	0.922612i	$-0.626050\pi$
$593$	−18.7863	−0.771460	−0.385730	+	0.922612i	$0.626050\pi$
$594$	0	0
$595$	−6.16229	−0.252629
$596$	0	0
$597$	−7.64602	−0.312931
$598$	0	0
$599$	27.4579	1.12190	0.560950	−	0.827850i	$-0.310436\pi$
$599$	27.4579	1.12190	0.560950	+	0.827850i	$0.310436\pi$
$600$	0	0
$601$	3.60426	0.147021	0.0735105	−	0.997294i	$-0.476580\pi$
$601$	3.60426	0.147021	0.0735105	+	0.997294i	$0.476580\pi$
$602$	0	0
$603$	19.0924	0.777501
$604$	0	0
$605$	12.8324	0.521709
$606$	0	0
$607$	−23.2924	−0.945410	−0.472705	−	0.881221i	$-0.656722\pi$
$607$	−23.2924	−0.945410	−0.472705	+	0.881221i	$0.656722\pi$
$608$	0	0
$609$	−0.657012	−0.0266235
$610$	0	0
$611$	6.01022	0.243147
$612$	0	0
$613$	11.5942	0.468284	0.234142	−	0.972202i	$-0.424772\pi$
$613$	11.5942	0.468284	0.234142	+	0.972202i	$0.424772\pi$
$614$	0	0
$615$	7.68893	0.310048
$616$	0	0
$617$	−18.3176	−0.737438	−0.368719	−	0.929541i	$-0.620204\pi$
$617$	−18.3176	−0.737438	−0.368719	+	0.929541i	$0.620204\pi$
$618$	0	0
$619$	23.9145	0.961206	0.480603	−	0.876938i	$-0.340418\pi$
$619$	23.9145	0.961206	0.480603	+	0.876938i	$0.340418\pi$
$620$	0	0
$621$	−21.7215	−0.871655
$622$	0	0
$623$	−0.237584	−0.00951860
$624$	0	0
$625$	−30.0646	−1.20259
$626$	0	0
$627$	−23.4041	−0.934668
$628$	0	0
$629$	−3.49481	−0.139347
$630$	0	0
$631$	−5.59809	−0.222857	−0.111428	−	0.993772i	$-0.535543\pi$
$631$	−5.59809	−0.222857	−0.111428	+	0.993772i	$0.535543\pi$
$632$	0	0
$633$	−14.3864	−0.571809
$634$	0	0
$635$	42.1887	1.67421
$636$	0	0
$637$	7.69508	0.304890
$638$	0	0
$639$	23.5576	0.931924
$640$	0	0
$641$	27.4040	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$641$	27.4040	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$642$	0	0
$643$	13.7877	0.543735	0.271868	−	0.962335i	$-0.412359\pi$
$643$	13.7877	0.543735	0.271868	+	0.962335i	$0.412359\pi$
$644$	0	0
$645$	7.01468	0.276203
$646$	0	0
$647$	−36.5366	−1.43640	−0.718200	−	0.695837i	$-0.755034\pi$
$647$	−36.5366	−1.43640	−0.718200	+	0.695837i	$0.755034\pi$
$648$	0	0
$649$	−55.2100	−2.16718
$650$	0	0
$651$	3.26395	0.127924
$652$	0	0
$653$	43.3663	1.69706	0.848528	−	0.529151i	$-0.177489\pi$
$653$	43.3663	1.69706	0.848528	+	0.529151i	$0.177489\pi$
$654$	0	0
$655$	23.5581	0.920490
$656$	0	0
$657$	−21.7809	−0.849752
$658$	0	0
$659$	2.56728	0.100007	0.0500035	−	0.998749i	$-0.484077\pi$
$659$	2.56728	0.100007	0.0500035	+	0.998749i	$0.484077\pi$
$660$	0	0
$661$	6.35014	0.246992	0.123496	−	0.992345i	$-0.460589\pi$
$661$	6.35014	0.246992	0.123496	+	0.992345i	$0.460589\pi$
$662$	0	0
$663$	8.44586	0.328010
$664$	0	0
$665$	4.75867	0.184533
$666$	0	0
$667$	8.08270	0.312964
$668$	0	0
$669$	11.6103	0.448879
$670$	0	0
$671$	−38.3988	−1.48237
$672$	0	0
$673$	−43.6943	−1.68429	−0.842147	−	0.539248i	$-0.818708\pi$
$673$	−43.6943	−1.68429	−0.842147	+	0.539248i	$0.818708\pi$
$674$	0	0
$675$	7.20520	0.277328
$676$	0	0
$677$	−26.6785	−1.02534	−0.512669	−	0.858586i	$-0.671343\pi$
$677$	−26.6785	−1.02534	−0.512669	+	0.858586i	$0.671343\pi$
$678$	0	0
$679$	−1.92156	−0.0737429
$680$	0	0
$681$	−27.6102	−1.05802
$682$	0	0
$683$	−36.1599	−1.38362	−0.691811	−	0.722079i	$-0.743187\pi$
$683$	−36.1599	−1.38362	−0.691811	+	0.722079i	$0.743187\pi$
$684$	0	0
$685$	−3.14633	−0.120215
$686$	0	0
$687$	−17.6734	−0.674282
$688$	0	0
$689$	−12.3706	−0.471284
$690$	0	0
$691$	23.9787	0.912191	0.456095	−	0.889931i	$-0.349248\pi$
$691$	23.9787	0.912191	0.456095	+	0.889931i	$0.349248\pi$
$692$	0	0
$693$	−2.57534	−0.0978289
$694$	0	0
$695$	6.98025	0.264776
$696$	0	0
$697$	−21.3697	−0.809436
$698$	0	0
$699$	17.2480	0.652379
$700$	0	0
$701$	21.5218	0.812866	0.406433	−	0.913681i	$-0.366772\pi$
$701$	21.5218	0.812866	0.406433	+	0.913681i	$0.366772\pi$
$702$	0	0
$703$	2.69878	0.101786
$704$	0	0
$705$	14.1210	0.531827
$706$	0	0
$707$	−5.15961	−0.194047
$708$	0	0
$709$	−48.1227	−1.80729	−0.903643	−	0.428287i	$-0.859117\pi$
$709$	−48.1227	−1.80729	−0.903643	+	0.428287i	$0.859117\pi$
$710$	0	0
$711$	−15.5933	−0.584794
$712$	0	0
$713$	−40.1538	−1.50377
$714$	0	0
$715$	11.3378	0.424010
$716$	0	0
$717$	−4.65441	−0.173822
$718$	0	0
$719$	−26.6998	−0.995736	−0.497868	−	0.867253i	$-0.665884\pi$
$719$	−26.6998	−0.995736	−0.497868	+	0.867253i	$0.665884\pi$
$720$	0	0
$721$	2.71712	0.101191
$722$	0	0
$723$	5.31501	0.197667
$724$	0	0
$725$	−2.68109	−0.0995733
$726$	0	0
$727$	45.3867	1.68330	0.841651	−	0.540023i	$-0.181584\pi$
$727$	45.3867	1.68330	0.841651	+	0.540023i	$0.181584\pi$
$728$	0	0
$729$	14.9426	0.553428
$730$	0	0
$731$	−19.4958	−0.721078
$732$	0	0
$733$	27.2436	1.00627	0.503133	−	0.864209i	$-0.332180\pi$
$733$	27.2436	1.00627	0.503133	+	0.864209i	$0.332180\pi$
$734$	0	0
$735$	18.0796	0.666876
$736$	0	0
$737$	39.7441	1.46399
$738$	0	0
$739$	−29.6122	−1.08930	−0.544651	−	0.838663i	$-0.683338\pi$
$739$	−29.6122	−1.08930	−0.544651	+	0.838663i	$0.683338\pi$
$740$	0	0
$741$	−6.52211	−0.239596
$742$	0	0
$743$	−5.94812	−0.218215	−0.109108	−	0.994030i	$-0.534799\pi$
$743$	−5.94812	−0.218215	−0.109108	+	0.994030i	$0.534799\pi$
$744$	0	0
$745$	41.3335	1.51434
$746$	0	0
$747$	−8.15669	−0.298438
$748$	0	0
$749$	−6.79534	−0.248296
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−22.4819	−0.819284
$754$	0	0
$755$	22.3466	0.813276
$756$	0	0
$757$	7.16070	0.260260	0.130130	−	0.991497i	$-0.458460\pi$
$757$	7.16070	0.260260	0.130130	+	0.991497i	$0.458460\pi$
$758$	0	0
$759$	−17.6771	−0.641638
$760$	0	0
$761$	8.61876	0.312430	0.156215	−	0.987723i	$-0.450071\pi$
$761$	8.61876	0.312430	0.156215	+	0.987723i	$0.450071\pi$
$762$	0	0
$763$	−2.73378	−0.0989695
$764$	0	0
$765$	−35.5652	−1.28586
$766$	0	0
$767$	−15.3856	−0.555542
$768$	0	0
$769$	22.7333	0.819785	0.409892	−	0.912134i	$-0.365566\pi$
$769$	22.7333	0.819785	0.409892	+	0.912134i	$0.365566\pi$
$770$	0	0
$771$	13.2439	0.476968
$772$	0	0
$773$	22.2371	0.799812	0.399906	−	0.916556i	$-0.369043\pi$
$773$	22.2371	0.799812	0.399906	+	0.916556i	$0.369043\pi$
$774$	0	0
$775$	13.3193	0.478445
$776$	0	0
$777$	−0.165693	−0.00594420
$778$	0	0
$779$	16.5022	0.591254
$780$	0	0
$781$	49.0392	1.75476
$782$	0	0
$783$	−9.69948	−0.346631
$784$	0	0
$785$	−24.5931	−0.877764
$786$	0	0
$787$	30.5341	1.08842	0.544212	−	0.838948i	$-0.316829\pi$
$787$	30.5341	1.08842	0.544212	+	0.838948i	$0.316829\pi$
$788$	0	0
$789$	4.17673	0.148696
$790$	0	0
$791$	−0.942457	−0.0335099
$792$	0	0
$793$	−10.7008	−0.379995
$794$	0	0
$795$	−29.0648	−1.03082
$796$	0	0
$797$	14.0680	0.498314	0.249157	−	0.968463i	$-0.419846\pi$
$797$	14.0680	0.498314	0.249157	+	0.968463i	$0.419846\pi$
$798$	0	0
$799$	−39.2463	−1.38843
$800$	0	0
$801$	−1.37120	−0.0484489
$802$	0	0
$803$	−45.3406	−1.60004
$804$	0	0
$805$	3.59423	0.126680
$806$	0	0
$807$	22.6537	0.797448
$808$	0	0
$809$	−6.60709	−0.232293	−0.116147	−	0.993232i	$-0.537054\pi$
$809$	−6.60709	−0.232293	−0.116147	+	0.993232i	$0.537054\pi$
$810$	0	0
$811$	−8.83123	−0.310106	−0.155053	−	0.987906i	$-0.549555\pi$
$811$	−8.83123	−0.310106	−0.155053	+	0.987906i	$0.549555\pi$
$812$	0	0
$813$	6.83002	0.239539
$814$	0	0
$815$	−47.5412	−1.66530
$816$	0	0
$817$	15.0551	0.526713
$818$	0	0
$819$	−0.717679	−0.0250777
$820$	0	0
$821$	−33.7581	−1.17816	−0.589082	−	0.808073i	$-0.700511\pi$
$821$	−33.7581	−1.17816	−0.589082	+	0.808073i	$0.700511\pi$
$822$	0	0
$823$	28.8890	1.00701	0.503504	−	0.863993i	$-0.332044\pi$
$823$	28.8890	1.00701	0.503504	+	0.863993i	$0.332044\pi$
$824$	0	0
$825$	5.86363	0.204145
$826$	0	0
$827$	29.2028	1.01548	0.507740	−	0.861511i	$-0.330481\pi$
$827$	29.2028	1.01548	0.507740	+	0.861511i	$0.330481\pi$
$828$	0	0
$829$	33.0765	1.14879	0.574396	−	0.818577i	$-0.305237\pi$
$829$	33.0765	1.14879	0.574396	+	0.818577i	$0.305237\pi$
$830$	0	0
$831$	−19.3864	−0.672507
$832$	0	0
$833$	−50.2483	−1.74100
$834$	0	0
$835$	−61.8174	−2.13928
$836$	0	0
$837$	48.1858	1.66554
$838$	0	0
$839$	25.6450	0.885362	0.442681	−	0.896679i	$-0.354027\pi$
$839$	25.6450	0.885362	0.442681	+	0.896679i	$0.354027\pi$
$840$	0	0
$841$	−25.3908	−0.875544
$842$	0	0
$843$	−23.0002	−0.792170
$844$	0	0
$845$	−29.7570	−1.02367
$846$	0	0
$847$	−1.69091	−0.0581003
$848$	0	0
$849$	11.6188	0.398757
$850$	0	0
$851$	2.03839	0.0698751
$852$	0	0
$853$	−14.8656	−0.508988	−0.254494	−	0.967074i	$-0.581909\pi$
$853$	−14.8656	−0.508988	−0.254494	+	0.967074i	$0.581909\pi$
$854$	0	0
$855$	27.4643	0.939261
$856$	0	0
$857$	3.29898	0.112691	0.0563455	−	0.998411i	$-0.482055\pi$
$857$	3.29898	0.112691	0.0563455	+	0.998411i	$0.482055\pi$
$858$	0	0
$859$	−39.6126	−1.35157	−0.675783	−	0.737101i	$-0.736194\pi$
$859$	−39.6126	−1.35157	−0.675783	+	0.737101i	$0.736194\pi$
$860$	0	0
$861$	−1.01316	−0.0345285
$862$	0	0
$863$	−18.4041	−0.626481	−0.313241	−	0.949674i	$-0.601415\pi$
$863$	−18.4041	−0.626481	−0.313241	+	0.949674i	$0.601415\pi$
$864$	0	0
$865$	−28.8247	−0.980068
$866$	0	0
$867$	−37.5299	−1.27458
$868$	0	0
$869$	−32.4601	−1.10113
$870$	0	0
$871$	11.0756	0.375284
$872$	0	0
$873$	−11.0902	−0.375345
$874$	0	0
$875$	3.03179	0.102493
$876$	0	0
$877$	−40.4029	−1.36431	−0.682154	−	0.731208i	$-0.738957\pi$
$877$	−40.4029	−1.36431	−0.682154	+	0.731208i	$0.738957\pi$
$878$	0	0
$879$	−13.3524	−0.450365
$880$	0	0
$881$	−31.4302	−1.05891	−0.529455	−	0.848338i	$-0.677603\pi$
$881$	−31.4302	−1.05891	−0.529455	+	0.848338i	$0.677603\pi$
$882$	0	0
$883$	11.3012	0.380316	0.190158	−	0.981753i	$-0.439100\pi$
$883$	11.3012	0.380316	0.190158	+	0.981753i	$0.439100\pi$
$884$	0	0
$885$	−36.1485	−1.21512
$886$	0	0
$887$	−25.1227	−0.843536	−0.421768	−	0.906704i	$-0.638590\pi$
$887$	−25.1227	−0.843536	−0.421768	+	0.906704i	$0.638590\pi$
$888$	0	0
$889$	−5.55917	−0.186449
$890$	0	0
$891$	−1.94331	−0.0651033
$892$	0	0
$893$	30.3070	1.01418
$894$	0	0
$895$	−8.58661	−0.287019
$896$	0	0
$897$	−4.92615	−0.164479
$898$	0	0
$899$	−17.9302	−0.598005
$900$	0	0
$901$	80.7794	2.69115
$902$	0	0
$903$	−0.924319	−0.0307594
$904$	0	0
$905$	−41.9236	−1.39359
$906$	0	0
$907$	−23.0128	−0.764127	−0.382063	−	0.924136i	$-0.624786\pi$
$907$	−23.0128	−0.764127	−0.382063	+	0.924136i	$0.624786\pi$
$908$	0	0
$909$	−29.7783	−0.987684
$910$	0	0
$911$	9.07981	0.300828	0.150414	−	0.988623i	$-0.451939\pi$
$911$	9.07981	0.300828	0.150414	+	0.988623i	$0.451939\pi$
$912$	0	0
$913$	−16.9796	−0.561942
$914$	0	0
$915$	−25.1414	−0.831150
$916$	0	0
$917$	−3.10423	−0.102511
$918$	0	0
$919$	−27.4842	−0.906621	−0.453310	−	0.891353i	$-0.649757\pi$
$919$	−27.4842	−0.906621	−0.453310	+	0.891353i	$0.649757\pi$
$920$	0	0
$921$	−20.1122	−0.662721
$922$	0	0
$923$	13.6660	0.449820
$924$	0	0
$925$	−0.676149	−0.0222317
$926$	0	0
$927$	15.6816	0.515053
$928$	0	0
$929$	53.6788	1.76115	0.880573	−	0.473911i	$-0.157158\pi$
$929$	53.6788	1.76115	0.880573	+	0.473911i	$0.157158\pi$
$930$	0	0
$931$	38.8030	1.27172
$932$	0	0
$933$	−21.3833	−0.700057
$934$	0	0
$935$	−74.0351	−2.42121
$936$	0	0
$937$	4.32473	0.141283	0.0706414	−	0.997502i	$-0.477495\pi$
$937$	4.32473	0.141283	0.0706414	+	0.997502i	$0.477495\pi$
$938$	0	0
$939$	−24.6561	−0.804621
$940$	0	0
$941$	−21.3622	−0.696386	−0.348193	−	0.937423i	$-0.613205\pi$
$941$	−21.3622	−0.696386	−0.348193	+	0.937423i	$0.613205\pi$
$942$	0	0
$943$	12.4642	0.405889
$944$	0	0
$945$	−4.31318	−0.140308
$946$	0	0
$947$	−26.8998	−0.874126	−0.437063	−	0.899431i	$-0.643981\pi$
$947$	−26.8998	−0.874126	−0.437063	+	0.899431i	$0.643981\pi$
$948$	0	0
$949$	−12.6353	−0.410158
$950$	0	0
$951$	−28.4142	−0.921395
$952$	0	0
$953$	−56.9505	−1.84481	−0.922404	−	0.386228i	$-0.873778\pi$
$953$	−56.9505	−1.84481	−0.922404	+	0.386228i	$0.873778\pi$
$954$	0	0
$955$	27.6490	0.894699
$956$	0	0
$957$	−7.89349	−0.255160
$958$	0	0
$959$	0.414589	0.0133878
$960$	0	0
$961$	58.0749	1.87338
$962$	0	0
$963$	−39.2188	−1.26381
$964$	0	0
$965$	36.3428	1.16991
$966$	0	0
$967$	−20.2950	−0.652642	−0.326321	−	0.945259i	$-0.605809\pi$
$967$	−20.2950	−0.652642	−0.326321	+	0.945259i	$0.605809\pi$
$968$	0	0
$969$	42.5889	1.36815
$970$	0	0
$971$	−28.2148	−0.905457	−0.452728	−	0.891649i	$-0.649549\pi$
$971$	−28.2148	−0.905457	−0.452728	+	0.891649i	$0.649549\pi$
$972$	0	0
$973$	−0.919781	−0.0294868
$974$	0	0
$975$	1.63404	0.0523312
$976$	0	0
$977$	6.06986	0.194192	0.0970961	−	0.995275i	$-0.469045\pi$
$977$	6.06986	0.194192	0.0970961	+	0.995275i	$0.469045\pi$
$978$	0	0
$979$	−2.85439	−0.0912266
$980$	0	0
$981$	−15.7778	−0.503747
$982$	0	0
$983$	17.6903	0.564233	0.282116	−	0.959380i	$-0.408964\pi$
$983$	17.6903	0.564233	0.282116	+	0.959380i	$0.408964\pi$
$984$	0	0
$985$	3.55184	0.113171
$986$	0	0
$987$	−1.86071	−0.0592271
$988$	0	0
$989$	11.3712	0.361582
$990$	0	0
$991$	−0.428154	−0.0136008	−0.00680038	−	0.999977i	$-0.502165\pi$
$991$	−0.428154	−0.0136008	−0.00680038	+	0.999977i	$0.502165\pi$
$992$	0	0
$993$	22.1641	0.703356
$994$	0	0
$995$	18.6778	0.592126
$996$	0	0
$997$	1.76540	0.0559109	0.0279554	−	0.999609i	$-0.491100\pi$
$997$	1.76540	0.0559109	0.0279554	+	0.999609i	$0.491100\pi$
$998$	0	0
$999$	−2.44613	−0.0773920

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.17	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.17	✓	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)

\(f(q)\)	\(=\)	\(q-1.03653 q^{3} +2.53205 q^{5} -0.333645 q^{7} -1.92561 q^{9} +O(q^{10})\)	\(q-1.03653 q^{3} +2.53205 q^{5} -0.333645 q^{7} -1.92561 q^{9} -4.00849 q^{11} -1.11706 q^{13} -2.62454 q^{15} +7.29433 q^{17} -5.63287 q^{19} +0.345833 q^{21} -4.25451 q^{23} +1.41125 q^{25} +5.10553 q^{27} -1.89980 q^{29} +9.43795 q^{31} +4.15491 q^{33} -0.844805 q^{35} -0.479113 q^{37} +1.15787 q^{39} -2.92963 q^{41} -2.67273 q^{43} -4.87573 q^{45} -5.38038 q^{47} -6.88868 q^{49} -7.56078 q^{51} +11.0743 q^{53} -10.1497 q^{55} +5.83863 q^{57} +13.7733 q^{59} +9.57938 q^{61} +0.642470 q^{63} -2.82845 q^{65} -9.91497 q^{67} +4.40992 q^{69} -12.2338 q^{71} +11.3112 q^{73} -1.46280 q^{75} +1.33741 q^{77} +8.09784 q^{79} +0.484799 q^{81} +4.23590 q^{83} +18.4696 q^{85} +1.96919 q^{87} +0.712086 q^{89} +0.372702 q^{91} -9.78270 q^{93} -14.2627 q^{95} +5.75930 q^{97} +7.71878 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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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