LMFDB - Embedded newform 6008.2.a.e.1.16

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			1.16
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)

$f(q)$	$=$	$q-1.33859 q^{3} +3.94917 q^{5} +0.220759 q^{7} -1.20816 q^{9} +O(q^{10})$	$q-1.33859 q^{3} +3.94917 q^{5} +0.220759 q^{7} -1.20816 q^{9} +1.87126 q^{11} +3.17517 q^{13} -5.28634 q^{15} +7.01370 q^{17} +5.00513 q^{19} -0.295507 q^{21} +1.70287 q^{23} +10.5960 q^{25} +5.63303 q^{27} +1.30440 q^{29} -4.27484 q^{31} -2.50486 q^{33} +0.871815 q^{35} +3.57462 q^{37} -4.25027 q^{39} +4.53410 q^{41} -3.45578 q^{43} -4.77125 q^{45} +1.75027 q^{47} -6.95127 q^{49} -9.38851 q^{51} -6.18967 q^{53} +7.38994 q^{55} -6.69984 q^{57} -10.8041 q^{59} +8.37748 q^{61} -0.266713 q^{63} +12.5393 q^{65} -3.87632 q^{67} -2.27945 q^{69} +3.47020 q^{71} -6.72897 q^{73} -14.1837 q^{75} +0.413098 q^{77} -6.08611 q^{79} -3.91584 q^{81} -6.03473 q^{83} +27.6983 q^{85} -1.74607 q^{87} +17.0339 q^{89} +0.700948 q^{91} +5.72227 q^{93} +19.7661 q^{95} +0.167792 q^{97} -2.26079 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.33859	−0.772838	−0.386419	−	0.922323i	$-0.626288\pi$
$3$	−1.33859	−0.772838	−0.386419	+	0.922323i	$0.626288\pi$
$4$	0	0
$5$	3.94917	1.76612	0.883062	−	0.469257i	$-0.155478\pi$
$5$	3.94917	1.76612	0.883062	+	0.469257i	$0.155478\pi$
$6$	0	0
$7$	0.220759	0.0834391	0.0417195	−	0.999129i	$-0.486716\pi$
$7$	0.220759	0.0834391	0.0417195	+	0.999129i	$0.486716\pi$
$8$	0	0
$9$	−1.20816	−0.402722
$10$	0	0
$11$	1.87126	0.564207	0.282104	−	0.959384i	$-0.408968\pi$
$11$	1.87126	0.564207	0.282104	+	0.959384i	$0.408968\pi$
$12$	0	0
$13$	3.17517	0.880635	0.440317	−	0.897842i	$-0.354866\pi$
$13$	3.17517	0.880635	0.440317	+	0.897842i	$0.354866\pi$
$14$	0	0
$15$	−5.28634	−1.36493
$16$	0	0
$17$	7.01370	1.70107	0.850537	−	0.525916i	$-0.176277\pi$
$17$	7.01370	1.70107	0.850537	+	0.525916i	$0.176277\pi$
$18$	0	0
$19$	5.00513	1.14826	0.574128	−	0.818766i	$-0.305341\pi$
$19$	5.00513	1.14826	0.574128	+	0.818766i	$0.305341\pi$
$20$	0	0
$21$	−0.295507	−0.0644849
$22$	0	0
$23$	1.70287	0.355073	0.177536	−	0.984114i	$-0.443187\pi$
$23$	1.70287	0.355073	0.177536	+	0.984114i	$0.443187\pi$
$24$	0	0
$25$	10.5960	2.11919
$26$	0	0
$27$	5.63303	1.08408
$28$	0	0
$29$	1.30440	0.242222	0.121111	−	0.992639i	$-0.461354\pi$
$29$	1.30440	0.242222	0.121111	+	0.992639i	$0.461354\pi$
$30$	0	0
$31$	−4.27484	−0.767783	−0.383892	−	0.923378i	$-0.625416\pi$
$31$	−4.27484	−0.767783	−0.383892	+	0.923378i	$0.625416\pi$
$32$	0	0
$33$	−2.50486	−0.436041
$34$	0	0
$35$	0.871815	0.147364
$36$	0	0
$37$	3.57462	0.587664	0.293832	−	0.955857i	$-0.405069\pi$
$37$	3.57462	0.587664	0.293832	+	0.955857i	$0.405069\pi$
$38$	0	0
$39$	−4.25027	−0.680588
$40$	0	0
$41$	4.53410	0.708107	0.354054	−	0.935225i	$-0.384803\pi$
$41$	4.53410	0.708107	0.354054	+	0.935225i	$0.384803\pi$
$42$	0	0
$43$	−3.45578	−0.527002	−0.263501	−	0.964659i	$-0.584877\pi$
$43$	−3.45578	−0.527002	−0.263501	+	0.964659i	$0.584877\pi$
$44$	0	0
$45$	−4.77125	−0.711256
$46$	0	0
$47$	1.75027	0.255302	0.127651	−	0.991819i	$-0.459256\pi$
$47$	1.75027	0.255302	0.127651	+	0.991819i	$0.459256\pi$
$48$	0	0
$49$	−6.95127	−0.993038
$50$	0	0
$51$	−9.38851	−1.31465
$52$	0	0
$53$	−6.18967	−0.850216	−0.425108	−	0.905143i	$-0.639764\pi$
$53$	−6.18967	−0.850216	−0.425108	+	0.905143i	$0.639764\pi$
$54$	0	0
$55$	7.38994	0.996460
$56$	0	0
$57$	−6.69984	−0.887416
$58$	0	0
$59$	−10.8041	−1.40658	−0.703290	−	0.710903i	$-0.748287\pi$
$59$	−10.8041	−1.40658	−0.703290	+	0.710903i	$0.748287\pi$
$60$	0	0
$61$	8.37748	1.07263	0.536313	−	0.844019i	$-0.319817\pi$
$61$	8.37748	1.07263	0.536313	+	0.844019i	$0.319817\pi$
$62$	0	0
$63$	−0.266713	−0.0336027
$64$	0	0
$65$	12.5393	1.55531
$66$	0	0
$67$	−3.87632	−0.473567	−0.236784	−	0.971562i	$-0.576093\pi$
$67$	−3.87632	−0.473567	−0.236784	+	0.971562i	$0.576093\pi$
$68$	0	0
$69$	−2.27945	−0.274414
$70$	0	0
$71$	3.47020	0.411837	0.205918	−	0.978569i	$-0.433982\pi$
$71$	3.47020	0.411837	0.205918	+	0.978569i	$0.433982\pi$
$72$	0	0
$73$	−6.72897	−0.787566	−0.393783	−	0.919203i	$-0.628834\pi$
$73$	−6.72897	−0.787566	−0.393783	+	0.919203i	$0.628834\pi$
$74$	0	0
$75$	−14.1837	−1.63779
$76$	0	0
$77$	0.413098	0.0470769
$78$	0	0
$79$	−6.08611	−0.684741	−0.342371	−	0.939565i	$-0.611230\pi$
$79$	−6.08611	−0.684741	−0.342371	+	0.939565i	$0.611230\pi$
$80$	0	0
$81$	−3.91584	−0.435094
$82$	0	0
$83$	−6.03473	−0.662398	−0.331199	−	0.943561i	$-0.607453\pi$
$83$	−6.03473	−0.662398	−0.331199	+	0.943561i	$0.607453\pi$
$84$	0	0
$85$	27.6983	3.00431
$86$	0	0
$87$	−1.74607	−0.187198
$88$	0	0
$89$	17.0339	1.80559	0.902794	−	0.430073i	$-0.141512\pi$
$89$	17.0339	1.80559	0.902794	+	0.430073i	$0.141512\pi$
$90$	0	0
$91$	0.700948	0.0734793
$92$	0	0
$93$	5.72227	0.593372
$94$	0	0
$95$	19.7661	2.02796
$96$	0	0
$97$	0.167792	0.0170367	0.00851834	−	0.999964i	$-0.497288\pi$
$97$	0.167792	0.0170367	0.00851834	+	0.999964i	$0.497288\pi$
$98$	0	0
$99$	−2.26079	−0.227218
$100$	0	0
$101$	−3.99967	−0.397982	−0.198991	−	0.980001i	$-0.563766\pi$
$101$	−3.99967	−0.397982	−0.198991	+	0.980001i	$0.563766\pi$
$102$	0	0
$103$	−3.38265	−0.333303	−0.166651	−	0.986016i	$-0.553295\pi$
$103$	−3.38265	−0.333303	−0.166651	+	0.986016i	$0.553295\pi$
$104$	0	0
$105$	−1.16701	−0.113888
$106$	0	0
$107$	−5.71506	−0.552496	−0.276248	−	0.961086i	$-0.589091\pi$
$107$	−5.71506	−0.552496	−0.276248	+	0.961086i	$0.589091\pi$
$108$	0	0
$109$	4.68816	0.449045	0.224522	−	0.974469i	$-0.427918\pi$
$109$	4.68816	0.449045	0.224522	+	0.974469i	$0.427918\pi$
$110$	0	0
$111$	−4.78497	−0.454169
$112$	0	0
$113$	−9.49848	−0.893542	−0.446771	−	0.894648i	$-0.647426\pi$
$113$	−9.49848	−0.893542	−0.446771	+	0.894648i	$0.647426\pi$
$114$	0	0
$115$	6.72492	0.627102
$116$	0	0
$117$	−3.83613	−0.354651
$118$	0	0
$119$	1.54834	0.141936
$120$	0	0
$121$	−7.49837	−0.681670
$122$	0	0
$123$	−6.06932	−0.547252
$124$	0	0
$125$	22.0994	1.97663
$126$	0	0
$127$	11.2302	0.996522	0.498261	−	0.867027i	$-0.333972\pi$
$127$	11.2302	0.996522	0.498261	+	0.867027i	$0.333972\pi$
$128$	0	0
$129$	4.62589	0.407287
$130$	0	0
$131$	14.3712	1.25562	0.627810	−	0.778367i	$-0.283952\pi$
$131$	14.3712	1.25562	0.627810	+	0.778367i	$0.283952\pi$
$132$	0	0
$133$	1.10493	0.0958094
$134$	0	0
$135$	22.2458	1.91461
$136$	0	0
$137$	9.33336	0.797403	0.398701	−	0.917081i	$-0.369461\pi$
$137$	9.33336	0.797403	0.398701	+	0.917081i	$0.369461\pi$
$138$	0	0
$139$	7.70690	0.653690	0.326845	−	0.945078i	$-0.394014\pi$
$139$	7.70690	0.653690	0.326845	+	0.945078i	$0.394014\pi$
$140$	0	0
$141$	−2.34290	−0.197307
$142$	0	0
$143$	5.94159	0.496860
$144$	0	0
$145$	5.15131	0.427793
$146$	0	0
$147$	9.30493	0.767457
$148$	0	0
$149$	−3.02064	−0.247460	−0.123730	−	0.992316i	$-0.539486\pi$
$149$	−3.02064	−0.247460	−0.123730	+	0.992316i	$0.539486\pi$
$150$	0	0
$151$	−2.76483	−0.224999	−0.112499	−	0.993652i	$-0.535886\pi$
$151$	−2.76483	−0.224999	−0.112499	+	0.993652i	$0.535886\pi$
$152$	0	0
$153$	−8.47371	−0.685059
$154$	0	0
$155$	−16.8821	−1.35600
$156$	0	0
$157$	−13.8352	−1.10417	−0.552083	−	0.833789i	$-0.686167\pi$
$157$	−13.8352	−1.10417	−0.552083	+	0.833789i	$0.686167\pi$
$158$	0	0
$159$	8.28545	0.657079
$160$	0	0
$161$	0.375923	0.0296269
$162$	0	0
$163$	−16.7370	−1.31094	−0.655472	−	0.755220i	$-0.727530\pi$
$163$	−16.7370	−1.31094	−0.655472	+	0.755220i	$0.727530\pi$
$164$	0	0
$165$	−9.89214	−0.770102
$166$	0	0
$167$	−10.6390	−0.823269	−0.411635	−	0.911349i	$-0.635042\pi$
$167$	−10.6390	−0.823269	−0.411635	+	0.911349i	$0.635042\pi$
$168$	0	0
$169$	−2.91827	−0.224482
$170$	0	0
$171$	−6.04702	−0.462427
$172$	0	0
$173$	−2.15546	−0.163877	−0.0819385	−	0.996637i	$-0.526111\pi$
$173$	−2.15546	−0.163877	−0.0819385	+	0.996637i	$0.526111\pi$
$174$	0	0
$175$	2.33915	0.176823
$176$	0	0
$177$	14.4624	1.08706
$178$	0	0
$179$	−19.6925	−1.47189	−0.735945	−	0.677041i	$-0.763262\pi$
$179$	−19.6925	−1.47189	−0.735945	+	0.677041i	$0.763262\pi$
$180$	0	0
$181$	17.5173	1.30205	0.651027	−	0.759054i	$-0.274338\pi$
$181$	17.5173	1.30205	0.651027	+	0.759054i	$0.274338\pi$
$182$	0	0
$183$	−11.2140	−0.828966
$184$	0	0
$185$	14.1168	1.03789
$186$	0	0
$187$	13.1245	0.959758
$188$	0	0
$189$	1.24354	0.0904543
$190$	0	0
$191$	10.5635	0.764345	0.382173	−	0.924091i	$-0.375176\pi$
$191$	10.5635	0.764345	0.382173	+	0.924091i	$0.375176\pi$
$192$	0	0
$193$	11.9677	0.861454	0.430727	−	0.902482i	$-0.358257\pi$
$193$	11.9677	0.861454	0.430727	+	0.902482i	$0.358257\pi$
$194$	0	0
$195$	−16.7850	−1.20200
$196$	0	0
$197$	−5.88879	−0.419559	−0.209779	−	0.977749i	$-0.567275\pi$
$197$	−5.88879	−0.419559	−0.209779	+	0.977749i	$0.567275\pi$
$198$	0	0
$199$	−13.9917	−0.991847	−0.495924	−	0.868366i	$-0.665170\pi$
$199$	−13.9917	−0.991847	−0.495924	+	0.868366i	$0.665170\pi$
$200$	0	0
$201$	5.18882	0.365991
$202$	0	0
$203$	0.287959	0.0202107
$204$	0	0
$205$	17.9059	1.25060
$206$	0	0
$207$	−2.05734	−0.142995
$208$	0	0
$209$	9.36592	0.647854
$210$	0	0
$211$	−9.47202	−0.652081	−0.326040	−	0.945356i	$-0.605715\pi$
$211$	−9.47202	−0.652081	−0.326040	+	0.945356i	$0.605715\pi$
$212$	0	0
$213$	−4.64519	−0.318283
$214$	0	0
$215$	−13.6475	−0.930750
$216$	0	0
$217$	−0.943709	−0.0640631
$218$	0	0
$219$	9.00736	0.608661
$220$	0	0
$221$	22.2697	1.49802
$222$	0	0
$223$	2.02819	0.135817	0.0679087	−	0.997692i	$-0.478367\pi$
$223$	2.02819	0.135817	0.0679087	+	0.997692i	$0.478367\pi$
$224$	0	0
$225$	−12.8017	−0.853444
$226$	0	0
$227$	−9.68842	−0.643043	−0.321522	−	0.946902i	$-0.604194\pi$
$227$	−9.68842	−0.643043	−0.321522	+	0.946902i	$0.604194\pi$
$228$	0	0
$229$	−16.4929	−1.08988	−0.544940	−	0.838475i	$-0.683448\pi$
$229$	−16.4929	−1.08988	−0.544940	+	0.838475i	$0.683448\pi$
$230$	0	0
$231$	−0.552971	−0.0363828
$232$	0	0
$233$	4.43586	0.290603	0.145301	−	0.989387i	$-0.453585\pi$
$233$	4.43586	0.290603	0.145301	+	0.989387i	$0.453585\pi$
$234$	0	0
$235$	6.91210	0.450896
$236$	0	0
$237$	8.14684	0.529194
$238$	0	0
$239$	19.0023	1.22915	0.614577	−	0.788857i	$-0.289327\pi$
$239$	19.0023	1.22915	0.614577	+	0.788857i	$0.289327\pi$
$240$	0	0
$241$	9.84615	0.634246	0.317123	−	0.948384i	$-0.397283\pi$
$241$	9.84615	0.634246	0.317123	+	0.948384i	$0.397283\pi$
$242$	0	0
$243$	−11.6574	−0.747819
$244$	0	0
$245$	−27.4517	−1.75383
$246$	0	0
$247$	15.8922	1.01119
$248$	0	0
$249$	8.07806	0.511926
$250$	0	0
$251$	−0.204291	−0.0128948	−0.00644738	−	0.999979i	$-0.502052\pi$
$251$	−0.204291	−0.0128948	−0.00644738	+	0.999979i	$0.502052\pi$
$252$	0	0
$253$	3.18652	0.200334
$254$	0	0
$255$	−37.0768	−2.32184
$256$	0	0
$257$	11.3238	0.706359	0.353180	−	0.935556i	$-0.385100\pi$
$257$	11.3238	0.706359	0.353180	+	0.935556i	$0.385100\pi$
$258$	0	0
$259$	0.789130	0.0490341
$260$	0	0
$261$	−1.57593	−0.0975479
$262$	0	0
$263$	11.8634	0.731530	0.365765	−	0.930707i	$-0.380807\pi$
$263$	11.8634	0.731530	0.365765	+	0.930707i	$0.380807\pi$
$264$	0	0
$265$	−24.4441	−1.50159
$266$	0	0
$267$	−22.8015	−1.39543
$268$	0	0
$269$	−6.98712	−0.426012	−0.213006	−	0.977051i	$-0.568325\pi$
$269$	−6.98712	−0.426012	−0.213006	+	0.977051i	$0.568325\pi$
$270$	0	0
$271$	−18.9261	−1.14968	−0.574840	−	0.818266i	$-0.694936\pi$
$271$	−18.9261	−1.14968	−0.574840	+	0.818266i	$0.694936\pi$
$272$	0	0
$273$	−0.938285	−0.0567876
$274$	0	0
$275$	19.8278	1.19566
$276$	0	0
$277$	−25.8130	−1.55095	−0.775476	−	0.631377i	$-0.782490\pi$
$277$	−25.8130	−1.55095	−0.775476	+	0.631377i	$0.782490\pi$
$278$	0	0
$279$	5.16471	0.309203
$280$	0	0
$281$	1.42272	0.0848723	0.0424362	−	0.999099i	$-0.486488\pi$
$281$	1.42272	0.0848723	0.0424362	+	0.999099i	$0.486488\pi$
$282$	0	0
$283$	−2.61426	−0.155402	−0.0777009	−	0.996977i	$-0.524758\pi$
$283$	−2.61426	−0.155402	−0.0777009	+	0.996977i	$0.524758\pi$
$284$	0	0
$285$	−26.4588	−1.56729
$286$	0	0
$287$	1.00094	0.0590838
$288$	0	0
$289$	32.1921	1.89365
$290$	0	0
$291$	−0.224605	−0.0131666
$292$	0	0
$293$	11.9357	0.697293	0.348646	−	0.937254i	$-0.386641\pi$
$293$	11.9357	0.697293	0.348646	+	0.937254i	$0.386641\pi$
$294$	0	0
$295$	−42.6674	−2.48419
$296$	0	0
$297$	10.5409	0.611644
$298$	0	0
$299$	5.40690	0.312689
$300$	0	0
$301$	−0.762895	−0.0439725
$302$	0	0
$303$	5.35393	0.307575
$304$	0	0
$305$	33.0841	1.89439
$306$	0	0
$307$	−3.80063	−0.216914	−0.108457	−	0.994101i	$-0.534591\pi$
$307$	−3.80063	−0.216914	−0.108457	+	0.994101i	$0.534591\pi$
$308$	0	0
$309$	4.52800	0.257589
$310$	0	0
$311$	15.7353	0.892266	0.446133	−	0.894967i	$-0.352801\pi$
$311$	15.7353	0.892266	0.446133	+	0.894967i	$0.352801\pi$
$312$	0	0
$313$	−9.56109	−0.540425	−0.270212	−	0.962801i	$-0.587094\pi$
$313$	−9.56109	−0.540425	−0.270212	+	0.962801i	$0.587094\pi$
$314$	0	0
$315$	−1.05330	−0.0593465
$316$	0	0
$317$	14.6210	0.821198	0.410599	−	0.911816i	$-0.365320\pi$
$317$	14.6210	0.821198	0.410599	+	0.911816i	$0.365320\pi$
$318$	0	0
$319$	2.44088	0.136663
$320$	0	0
$321$	7.65015	0.426990
$322$	0	0
$323$	35.1045	1.95327
$324$	0	0
$325$	33.6440	1.86623
$326$	0	0
$327$	−6.27555	−0.347039
$328$	0	0
$329$	0.386387	0.0213022
$330$	0	0
$331$	11.0239	0.605927	0.302964	−	0.953002i	$-0.402024\pi$
$331$	11.0239	0.605927	0.302964	+	0.953002i	$0.402024\pi$
$332$	0	0
$333$	−4.31873	−0.236665
$334$	0	0
$335$	−15.3082	−0.836378
$336$	0	0
$337$	28.8377	1.57089	0.785444	−	0.618933i	$-0.212435\pi$
$337$	28.8377	1.57089	0.785444	+	0.618933i	$0.212435\pi$
$338$	0	0
$339$	12.7146	0.690563
$340$	0	0
$341$	−7.99935	−0.433189
$342$	0	0
$343$	−3.07987	−0.166297
$344$	0	0
$345$	−9.00194	−0.484648
$346$	0	0
$347$	32.9419	1.76842	0.884208	−	0.467093i	$-0.154699\pi$
$347$	32.9419	1.76842	0.884208	+	0.467093i	$0.154699\pi$
$348$	0	0
$349$	9.05627	0.484771	0.242385	−	0.970180i	$-0.422070\pi$
$349$	9.05627	0.484771	0.242385	+	0.970180i	$0.422070\pi$
$350$	0	0
$351$	17.8858	0.954675
$352$	0	0
$353$	−8.15793	−0.434203	−0.217101	−	0.976149i	$-0.569660\pi$
$353$	−8.15793	−0.434203	−0.217101	+	0.976149i	$0.569660\pi$
$354$	0	0
$355$	13.7044	0.727355
$356$	0	0
$357$	−2.07260	−0.109693
$358$	0	0
$359$	−19.1463	−1.01050	−0.505250	−	0.862973i	$-0.668600\pi$
$359$	−19.1463	−1.01050	−0.505250	+	0.862973i	$0.668600\pi$
$360$	0	0
$361$	6.05135	0.318492
$362$	0	0
$363$	10.0373	0.526821
$364$	0	0
$365$	−26.5739	−1.39094
$366$	0	0
$367$	−1.10651	−0.0577592	−0.0288796	−	0.999583i	$-0.509194\pi$
$367$	−1.10651	−0.0577592	−0.0288796	+	0.999583i	$0.509194\pi$
$368$	0	0
$369$	−5.47794	−0.285170
$370$	0	0
$371$	−1.36642	−0.0709412
$372$	0	0
$373$	5.10511	0.264333	0.132166	−	0.991228i	$-0.457807\pi$
$373$	5.10511	0.264333	0.132166	+	0.991228i	$0.457807\pi$
$374$	0	0
$375$	−29.5821	−1.52762
$376$	0	0
$377$	4.14171	0.213309
$378$	0	0
$379$	17.7155	0.909986	0.454993	−	0.890495i	$-0.349642\pi$
$379$	17.7155	0.909986	0.454993	+	0.890495i	$0.349642\pi$
$380$	0	0
$381$	−15.0327	−0.770150
$382$	0	0
$383$	−36.3290	−1.85632	−0.928161	−	0.372178i	$-0.878611\pi$
$383$	−36.3290	−1.85632	−0.928161	+	0.372178i	$0.878611\pi$
$384$	0	0
$385$	1.63140	0.0831436
$386$	0	0
$387$	4.17515	0.212235
$388$	0	0
$389$	12.3601	0.626684	0.313342	−	0.949640i	$-0.398551\pi$
$389$	12.3601	0.626684	0.313342	+	0.949640i	$0.398551\pi$
$390$	0	0
$391$	11.9434	0.604004
$392$	0	0
$393$	−19.2372	−0.970390
$394$	0	0
$395$	−24.0351	−1.20934
$396$	0	0
$397$	−11.4293	−0.573623	−0.286811	−	0.957987i	$-0.592595\pi$
$397$	−11.4293	−0.573623	−0.286811	+	0.957987i	$0.592595\pi$
$398$	0	0
$399$	−1.47905	−0.0740451
$400$	0	0
$401$	0.0721583	0.00360342	0.00180171	−	0.999998i	$-0.499426\pi$
$401$	0.0721583	0.00360342	0.00180171	+	0.999998i	$0.499426\pi$
$402$	0	0
$403$	−13.5733	−0.676137
$404$	0	0
$405$	−15.4643	−0.768429
$406$	0	0
$407$	6.68906	0.331564
$408$	0	0
$409$	11.1992	0.553766	0.276883	−	0.960904i	$-0.410699\pi$
$409$	11.1992	0.553766	0.276883	+	0.960904i	$0.410699\pi$
$410$	0	0
$411$	−12.4936	−0.616263
$412$	0	0
$413$	−2.38511	−0.117364
$414$	0	0
$415$	−23.8322	−1.16988
$416$	0	0
$417$	−10.3164	−0.505197
$418$	0	0
$419$	−9.70567	−0.474153	−0.237076	−	0.971491i	$-0.576189\pi$
$419$	−9.70567	−0.474153	−0.237076	+	0.971491i	$0.576189\pi$
$420$	0	0
$421$	7.09943	0.346005	0.173002	−	0.984921i	$-0.444653\pi$
$421$	7.09943	0.346005	0.173002	+	0.984921i	$0.444653\pi$
$422$	0	0
$423$	−2.11461	−0.102816
$424$	0	0
$425$	74.3169	3.60490
$426$	0	0
$427$	1.84940	0.0894989
$428$	0	0
$429$	−7.95338	−0.383993
$430$	0	0
$431$	23.0353	1.10957	0.554787	−	0.831993i	$-0.312800\pi$
$431$	23.0353	1.10957	0.554787	+	0.831993i	$0.312800\pi$
$432$	0	0
$433$	9.68490	0.465427	0.232713	−	0.972545i	$-0.425240\pi$
$433$	9.68490	0.465427	0.232713	+	0.972545i	$0.425240\pi$
$434$	0	0
$435$	−6.89552	−0.330615
$436$	0	0
$437$	8.52308	0.407714
$438$	0	0
$439$	19.4680	0.929156	0.464578	−	0.885532i	$-0.346206\pi$
$439$	19.4680	0.929156	0.464578	+	0.885532i	$0.346206\pi$
$440$	0	0
$441$	8.39827	0.399918
$442$	0	0
$443$	12.5356	0.595583	0.297791	−	0.954631i	$-0.403750\pi$
$443$	12.5356	0.595583	0.297791	+	0.954631i	$0.403750\pi$
$444$	0	0
$445$	67.2698	3.18889
$446$	0	0
$447$	4.04341	0.191247
$448$	0	0
$449$	8.76851	0.413812	0.206906	−	0.978361i	$-0.433661\pi$
$449$	8.76851	0.413812	0.206906	+	0.978361i	$0.433661\pi$
$450$	0	0
$451$	8.48450	0.399519
$452$	0	0
$453$	3.70099	0.173888
$454$	0	0
$455$	2.76816	0.129774
$456$	0	0
$457$	−23.9543	−1.12053	−0.560267	−	0.828312i	$-0.689301\pi$
$457$	−23.9543	−1.12053	−0.560267	+	0.828312i	$0.689301\pi$
$458$	0	0
$459$	39.5084	1.84409
$460$	0	0
$461$	−15.5606	−0.724731	−0.362366	−	0.932036i	$-0.618031\pi$
$461$	−15.5606	−0.724731	−0.362366	+	0.932036i	$0.618031\pi$
$462$	0	0
$463$	−22.4131	−1.04162	−0.520812	−	0.853671i	$-0.674371\pi$
$463$	−22.4131	−1.04162	−0.520812	+	0.853671i	$0.674371\pi$
$464$	0	0
$465$	22.5982	1.04797
$466$	0	0
$467$	36.1132	1.67112	0.835559	−	0.549401i	$-0.185144\pi$
$467$	36.1132	1.67112	0.835559	+	0.549401i	$0.185144\pi$
$468$	0	0
$469$	−0.855732	−0.0395140
$470$	0	0
$471$	18.5197	0.853341
$472$	0	0
$473$	−6.46668	−0.297338
$474$	0	0
$475$	53.0342	2.43337
$476$	0	0
$477$	7.47813	0.342400
$478$	0	0
$479$	−14.6880	−0.671114	−0.335557	−	0.942020i	$-0.608924\pi$
$479$	−14.6880	−0.671114	−0.335557	+	0.942020i	$0.608924\pi$
$480$	0	0
$481$	11.3500	0.517517
$482$	0	0
$483$	−0.503209	−0.0228968
$484$	0	0
$485$	0.662639	0.0300889
$486$	0	0
$487$	−5.44267	−0.246631	−0.123315	−	0.992368i	$-0.539353\pi$
$487$	−5.44267	−0.246631	−0.123315	+	0.992368i	$0.539353\pi$
$488$	0	0
$489$	22.4041	1.01315
$490$	0	0
$491$	0.113956	0.00514276	0.00257138	−	0.999997i	$-0.499182\pi$
$491$	0.113956	0.00514276	0.00257138	+	0.999997i	$0.499182\pi$
$492$	0	0
$493$	9.14870	0.412037
$494$	0	0
$495$	−8.92827	−0.401296
$496$	0	0
$497$	0.766078	0.0343633
$498$	0	0
$499$	−34.9379	−1.56404	−0.782018	−	0.623256i	$-0.785809\pi$
$499$	−34.9379	−1.56404	−0.782018	+	0.623256i	$0.785809\pi$
$500$	0	0
$501$	14.2413	0.636254
$502$	0	0
$503$	8.40784	0.374887	0.187444	−	0.982275i	$-0.439980\pi$
$503$	8.40784	0.374887	0.187444	+	0.982275i	$0.439980\pi$
$504$	0	0
$505$	−15.7954	−0.702885
$506$	0	0
$507$	3.90638	0.173489
$508$	0	0
$509$	15.2112	0.674225	0.337113	−	0.941464i	$-0.390550\pi$
$509$	15.2112	0.674225	0.337113	+	0.941464i	$0.390550\pi$
$510$	0	0
$511$	−1.48548	−0.0657138
$512$	0	0
$513$	28.1940	1.24480
$514$	0	0
$515$	−13.3587	−0.588654
$516$	0	0
$517$	3.27521	0.144044
$518$	0	0
$519$	2.88529	0.126650
$520$	0	0
$521$	−32.1593	−1.40893	−0.704463	−	0.709740i	$-0.748812\pi$
$521$	−32.1593	−1.40893	−0.704463	+	0.709740i	$0.748812\pi$
$522$	0	0
$523$	−3.96769	−0.173495	−0.0867476	−	0.996230i	$-0.527647\pi$
$523$	−3.96769	−0.173495	−0.0867476	+	0.996230i	$0.527647\pi$
$524$	0	0
$525$	−3.13118	−0.136656
$526$	0	0
$527$	−29.9824	−1.30606
$528$	0	0
$529$	−20.1002	−0.873923
$530$	0	0
$531$	13.0532	0.566460
$532$	0	0
$533$	14.3966	0.623584
$534$	0	0
$535$	−22.5698	−0.975776
$536$	0	0
$537$	26.3603	1.13753
$538$	0	0
$539$	−13.0077	−0.560279
$540$	0	0
$541$	−15.5362	−0.667955	−0.333978	−	0.942581i	$-0.608391\pi$
$541$	−15.5362	−0.667955	−0.333978	+	0.942581i	$0.608391\pi$
$542$	0	0
$543$	−23.4486	−1.00628
$544$	0	0
$545$	18.5144	0.793068
$546$	0	0
$547$	38.4936	1.64587	0.822934	−	0.568137i	$-0.192336\pi$
$547$	38.4936	1.64587	0.822934	+	0.568137i	$0.192336\pi$
$548$	0	0
$549$	−10.1214	−0.431970
$550$	0	0
$551$	6.52871	0.278132
$552$	0	0
$553$	−1.34356	−0.0571342
$554$	0	0
$555$	−18.8967	−0.802119
$556$	0	0
$557$	−15.3414	−0.650036	−0.325018	−	0.945708i	$-0.605370\pi$
$557$	−15.3414	−0.650036	−0.325018	+	0.945708i	$0.605370\pi$
$558$	0	0
$559$	−10.9727	−0.464096
$560$	0	0
$561$	−17.5684	−0.741737
$562$	0	0
$563$	41.1512	1.73432	0.867158	−	0.498034i	$-0.165944\pi$
$563$	41.1512	1.73432	0.867158	+	0.498034i	$0.165944\pi$
$564$	0	0
$565$	−37.5111	−1.57811
$566$	0	0
$567$	−0.864458	−0.0363038
$568$	0	0
$569$	−27.0512	−1.13404	−0.567022	−	0.823703i	$-0.691905\pi$
$569$	−27.0512	−1.13404	−0.567022	+	0.823703i	$0.691905\pi$
$570$	0	0
$571$	19.7641	0.827104	0.413552	−	0.910481i	$-0.364288\pi$
$571$	19.7641	0.827104	0.413552	+	0.910481i	$0.364288\pi$
$572$	0	0
$573$	−14.1402	−0.590715
$574$	0	0
$575$	18.0435	0.752467
$576$	0	0
$577$	−15.3608	−0.639477	−0.319739	−	0.947506i	$-0.603595\pi$
$577$	−15.3608	−0.639477	−0.319739	+	0.947506i	$0.603595\pi$
$578$	0	0
$579$	−16.0199	−0.665764
$580$	0	0
$581$	−1.33222	−0.0552699
$582$	0	0
$583$	−11.5825	−0.479698
$584$	0	0
$585$	−15.1495	−0.626357
$586$	0	0
$587$	15.7039	0.648169	0.324085	−	0.946028i	$-0.394944\pi$
$587$	15.7039	0.648169	0.324085	+	0.946028i	$0.394944\pi$
$588$	0	0
$589$	−21.3961	−0.881612
$590$	0	0
$591$	7.88270	0.324251
$592$	0	0
$593$	34.0904	1.39992	0.699962	−	0.714180i	$-0.253200\pi$
$593$	34.0904	1.39992	0.699962	+	0.714180i	$0.253200\pi$
$594$	0	0
$595$	6.11465	0.250676
$596$	0	0
$597$	18.7293	0.766537
$598$	0	0
$599$	6.54803	0.267545	0.133773	−	0.991012i	$-0.457291\pi$
$599$	6.54803	0.267545	0.133773	+	0.991012i	$0.457291\pi$
$600$	0	0
$601$	−26.8217	−1.09408	−0.547039	−	0.837107i	$-0.684245\pi$
$601$	−26.8217	−1.09408	−0.547039	+	0.837107i	$0.684245\pi$
$602$	0	0
$603$	4.68323	0.190716
$604$	0	0
$605$	−29.6124	−1.20391
$606$	0	0
$607$	25.8143	1.04777	0.523885	−	0.851789i	$-0.324482\pi$
$607$	25.8143	1.04777	0.523885	+	0.851789i	$0.324482\pi$
$608$	0	0
$609$	−0.385460	−0.0156196
$610$	0	0
$611$	5.55740	0.224828
$612$	0	0
$613$	3.54189	0.143056	0.0715278	−	0.997439i	$-0.477213\pi$
$613$	3.54189	0.143056	0.0715278	+	0.997439i	$0.477213\pi$
$614$	0	0
$615$	−23.9688	−0.966515
$616$	0	0
$617$	−9.07073	−0.365174	−0.182587	−	0.983190i	$-0.558447\pi$
$617$	−9.07073	−0.365174	−0.182587	+	0.983190i	$0.558447\pi$
$618$	0	0
$619$	−36.2957	−1.45885	−0.729423	−	0.684063i	$-0.760211\pi$
$619$	−36.2957	−1.45885	−0.729423	+	0.684063i	$0.760211\pi$
$620$	0	0
$621$	9.59230	0.384926
$622$	0	0
$623$	3.76038	0.150657
$624$	0	0
$625$	34.2946	1.37178
$626$	0	0
$627$	−12.5372	−0.500686
$628$	0	0
$629$	25.0713	0.999659
$630$	0	0
$631$	16.7823	0.668091	0.334046	−	0.942557i	$-0.391586\pi$
$631$	16.7823	0.668091	0.334046	+	0.942557i	$0.391586\pi$
$632$	0	0
$633$	12.6792	0.503953
$634$	0	0
$635$	44.3501	1.75998
$636$	0	0
$637$	−22.0715	−0.874504
$638$	0	0
$639$	−4.19257	−0.165856
$640$	0	0
$641$	25.9725	1.02585	0.512925	−	0.858433i	$-0.328562\pi$
$641$	25.9725	1.02585	0.512925	+	0.858433i	$0.328562\pi$
$642$	0	0
$643$	−21.0960	−0.831947	−0.415973	−	0.909377i	$-0.636559\pi$
$643$	−21.0960	−0.831947	−0.415973	+	0.909377i	$0.636559\pi$
$644$	0	0
$645$	18.2684	0.719319
$646$	0	0
$647$	5.25249	0.206497	0.103248	−	0.994656i	$-0.467076\pi$
$647$	5.25249	0.206497	0.103248	+	0.994656i	$0.467076\pi$
$648$	0	0
$649$	−20.2174	−0.793603
$650$	0	0
$651$	1.26324	0.0495104
$652$	0	0
$653$	−27.2221	−1.06528	−0.532642	−	0.846340i	$-0.678801\pi$
$653$	−27.2221	−1.06528	−0.532642	+	0.846340i	$0.678801\pi$
$654$	0	0
$655$	56.7544	2.21758
$656$	0	0
$657$	8.12970	0.317170
$658$	0	0
$659$	16.4501	0.640804	0.320402	−	0.947282i	$-0.396182\pi$
$659$	16.4501	0.640804	0.320402	+	0.947282i	$0.396182\pi$
$660$	0	0
$661$	−10.3556	−0.402787	−0.201393	−	0.979510i	$-0.564547\pi$
$661$	−10.3556	−0.402787	−0.201393	+	0.979510i	$0.564547\pi$
$662$	0	0
$663$	−29.8101	−1.15773
$664$	0	0
$665$	4.36355	0.169211
$666$	0	0
$667$	2.22123	0.0860063
$668$	0	0
$669$	−2.71492	−0.104965
$670$	0	0
$671$	15.6765	0.605183
$672$	0	0
$673$	0.912732	0.0351832	0.0175916	−	0.999845i	$-0.494400\pi$
$673$	0.912732	0.0351832	0.0175916	+	0.999845i	$0.494400\pi$
$674$	0	0
$675$	59.6873	2.29737
$676$	0	0
$677$	−13.3888	−0.514573	−0.257287	−	0.966335i	$-0.582828\pi$
$677$	−13.3888	−0.514573	−0.257287	+	0.966335i	$0.582828\pi$
$678$	0	0
$679$	0.0370416	0.00142152
$680$	0	0
$681$	12.9689	0.496968
$682$	0	0
$683$	−22.3662	−0.855818	−0.427909	−	0.903822i	$-0.640750\pi$
$683$	−22.3662	−0.855818	−0.427909	+	0.903822i	$0.640750\pi$
$684$	0	0
$685$	36.8591	1.40831
$686$	0	0
$687$	22.0773	0.842301
$688$	0	0
$689$	−19.6533	−0.748730
$690$	0	0
$691$	−15.4244	−0.586773	−0.293386	−	0.955994i	$-0.594782\pi$
$691$	−15.4244	−0.586773	−0.293386	+	0.955994i	$0.594782\pi$
$692$	0	0
$693$	−0.499091	−0.0189589
$694$	0	0
$695$	30.4359	1.15450
$696$	0	0
$697$	31.8008	1.20454
$698$	0	0
$699$	−5.93782	−0.224589
$700$	0	0
$701$	19.0396	0.719116	0.359558	−	0.933123i	$-0.382927\pi$
$701$	19.0396	0.719116	0.359558	+	0.933123i	$0.382927\pi$
$702$	0	0
$703$	17.8914	0.674789
$704$	0	0
$705$	−9.25250	−0.348469
$706$	0	0
$707$	−0.882963	−0.0332072
$708$	0	0
$709$	5.64775	0.212106	0.106053	−	0.994360i	$-0.466179\pi$
$709$	5.64775	0.212106	0.106053	+	0.994360i	$0.466179\pi$
$710$	0	0
$711$	7.35303	0.275760
$712$	0	0
$713$	−7.27948	−0.272619
$714$	0	0
$715$	23.4643	0.877517
$716$	0	0
$717$	−25.4363	−0.949937
$718$	0	0
$719$	−21.4226	−0.798927	−0.399463	−	0.916749i	$-0.630804\pi$
$719$	−21.4226	−0.798927	−0.399463	+	0.916749i	$0.630804\pi$
$720$	0	0
$721$	−0.746751	−0.0278105
$722$	0	0
$723$	−13.1800	−0.490170
$724$	0	0
$725$	13.8214	0.513314
$726$	0	0
$727$	13.2070	0.489821	0.244911	−	0.969546i	$-0.421241\pi$
$727$	13.2070	0.489821	0.244911	+	0.969546i	$0.421241\pi$
$728$	0	0
$729$	27.3520	1.01304
$730$	0	0
$731$	−24.2378	−0.896469
$732$	0	0
$733$	34.7367	1.28303	0.641515	−	0.767111i	$-0.278306\pi$
$733$	34.7367	1.28303	0.641515	+	0.767111i	$0.278306\pi$
$734$	0	0
$735$	36.7468	1.35542
$736$	0	0
$737$	−7.25361	−0.267190
$738$	0	0
$739$	37.1757	1.36753	0.683765	−	0.729702i	$-0.260341\pi$
$739$	37.1757	1.36753	0.683765	+	0.729702i	$0.260341\pi$
$740$	0	0
$741$	−21.2732	−0.781489
$742$	0	0
$743$	43.0654	1.57992	0.789959	−	0.613160i	$-0.210102\pi$
$743$	43.0654	1.57992	0.789959	+	0.613160i	$0.210102\pi$
$744$	0	0
$745$	−11.9290	−0.437046
$746$	0	0
$747$	7.29095	0.266762
$748$	0	0
$749$	−1.26165	−0.0460998
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	0.273463	0.00996556
$754$	0	0
$755$	−10.9188	−0.397376
$756$	0	0
$757$	−43.8148	−1.59248	−0.796238	−	0.604984i	$-0.793180\pi$
$757$	−43.8148	−1.59248	−0.796238	+	0.604984i	$0.793180\pi$
$758$	0	0
$759$	−4.26545	−0.154826
$760$	0	0
$761$	−1.72407	−0.0624975	−0.0312488	−	0.999512i	$-0.509948\pi$
$761$	−1.72407	−0.0624975	−0.0312488	+	0.999512i	$0.509948\pi$
$762$	0	0
$763$	1.03495	0.0374679
$764$	0	0
$765$	−33.4641	−1.20990
$766$	0	0
$767$	−34.3050	−1.23868
$768$	0	0
$769$	36.3950	1.31244	0.656218	−	0.754571i	$-0.272155\pi$
$769$	36.3950	1.31244	0.656218	+	0.754571i	$0.272155\pi$
$770$	0	0
$771$	−15.1580	−0.545901
$772$	0	0
$773$	6.21741	0.223625	0.111812	−	0.993729i	$-0.464334\pi$
$773$	6.21741	0.223625	0.111812	+	0.993729i	$0.464334\pi$
$774$	0	0
$775$	−45.2960	−1.62708
$776$	0	0
$777$	−1.05632	−0.0378954
$778$	0	0
$779$	22.6938	0.813089
$780$	0	0
$781$	6.49366	0.232361
$782$	0	0
$783$	7.34774	0.262587
$784$	0	0
$785$	−54.6374	−1.95009
$786$	0	0
$787$	−28.6165	−1.02007	−0.510034	−	0.860154i	$-0.670367\pi$
$787$	−28.6165	−1.02007	−0.510034	+	0.860154i	$0.670367\pi$
$788$	0	0
$789$	−15.8803	−0.565354
$790$	0	0
$791$	−2.09688	−0.0745563
$792$	0	0
$793$	26.5999	0.944592
$794$	0	0
$795$	32.7207	1.16048
$796$	0	0
$797$	51.0966	1.80994	0.904968	−	0.425480i	$-0.139895\pi$
$797$	51.0966	1.80994	0.904968	+	0.425480i	$0.139895\pi$
$798$	0	0
$799$	12.2758	0.434288
$800$	0	0
$801$	−20.5797	−0.727149
$802$	0	0
$803$	−12.5917	−0.444351
$804$	0	0
$805$	1.48459	0.0523248
$806$	0	0
$807$	9.35291	0.329238
$808$	0	0
$809$	−40.1193	−1.41052	−0.705259	−	0.708949i	$-0.749169\pi$
$809$	−40.1193	−1.41052	−0.705259	+	0.708949i	$0.749169\pi$
$810$	0	0
$811$	33.2049	1.16598	0.582991	−	0.812479i	$-0.301882\pi$
$811$	33.2049	1.16598	0.582991	+	0.812479i	$0.301882\pi$
$812$	0	0
$813$	25.3344	0.888516
$814$	0	0
$815$	−66.0973	−2.31529
$816$	0	0
$817$	−17.2966	−0.605133
$818$	0	0
$819$	−0.846861	−0.0295917
$820$	0	0
$821$	−0.533428	−0.0186168	−0.00930838	−	0.999957i	$-0.502963\pi$
$821$	−0.533428	−0.0186168	−0.00930838	+	0.999957i	$0.502963\pi$
$822$	0	0
$823$	−31.8473	−1.11013	−0.555063	−	0.831808i	$-0.687306\pi$
$823$	−31.8473	−1.11013	−0.555063	+	0.831808i	$0.687306\pi$
$824$	0	0
$825$	−26.5414	−0.924054
$826$	0	0
$827$	19.5264	0.678998	0.339499	−	0.940606i	$-0.389742\pi$
$827$	19.5264	0.678998	0.339499	+	0.940606i	$0.389742\pi$
$828$	0	0
$829$	−39.1148	−1.35851	−0.679257	−	0.733901i	$-0.737698\pi$
$829$	−39.1148	−1.35851	−0.679257	+	0.733901i	$0.737698\pi$
$830$	0	0
$831$	34.5531	1.19863
$832$	0	0
$833$	−48.7541	−1.68923
$834$	0	0
$835$	−42.0152	−1.45400
$836$	0	0
$837$	−24.0803	−0.832336
$838$	0	0
$839$	22.9760	0.793221	0.396610	−	0.917987i	$-0.370186\pi$
$839$	22.9760	0.793221	0.396610	+	0.917987i	$0.370186\pi$
$840$	0	0
$841$	−27.2985	−0.941329
$842$	0	0
$843$	−1.90444	−0.0655925
$844$	0	0
$845$	−11.5248	−0.396464
$846$	0	0
$847$	−1.65533	−0.0568779
$848$	0	0
$849$	3.49944	0.120100
$850$	0	0
$851$	6.08711	0.208663
$852$	0	0
$853$	54.2863	1.85873	0.929363	−	0.369166i	$-0.120357\pi$
$853$	54.2863	1.85873	0.929363	+	0.369166i	$0.120357\pi$
$854$	0	0
$855$	−23.8807	−0.816704
$856$	0	0
$857$	−45.7939	−1.56429	−0.782145	−	0.623096i	$-0.785875\pi$
$857$	−45.7939	−1.56429	−0.782145	+	0.623096i	$0.785875\pi$
$858$	0	0
$859$	−0.226434	−0.00772583	−0.00386291	−	0.999993i	$-0.501230\pi$
$859$	−0.226434	−0.00772583	−0.00386291	+	0.999993i	$0.501230\pi$
$860$	0	0
$861$	−1.33986	−0.0456622
$862$	0	0
$863$	48.7597	1.65980	0.829900	−	0.557912i	$-0.188397\pi$
$863$	48.7597	1.65980	0.829900	+	0.557912i	$0.188397\pi$
$864$	0	0
$865$	−8.51230	−0.289427
$866$	0	0
$867$	−43.0921	−1.46348
$868$	0	0
$869$	−11.3887	−0.386336
$870$	0	0
$871$	−12.3080	−0.417040
$872$	0	0
$873$	−0.202720	−0.00686104
$874$	0	0
$875$	4.87864	0.164928
$876$	0	0
$877$	19.3293	0.652706	0.326353	−	0.945248i	$-0.394180\pi$
$877$	19.3293	0.652706	0.326353	+	0.945248i	$0.394180\pi$
$878$	0	0
$879$	−15.9771	−0.538894
$880$	0	0
$881$	−48.6138	−1.63784	−0.818920	−	0.573908i	$-0.805427\pi$
$881$	−48.6138	−1.63784	−0.818920	+	0.573908i	$0.805427\pi$
$882$	0	0
$883$	−10.8725	−0.365888	−0.182944	−	0.983123i	$-0.558563\pi$
$883$	−10.8725	−0.365888	−0.182944	+	0.983123i	$0.558563\pi$
$884$	0	0
$885$	57.1144	1.91988
$886$	0	0
$887$	−24.5323	−0.823715	−0.411858	−	0.911248i	$-0.635120\pi$
$887$	−24.5323	−0.823715	−0.411858	+	0.911248i	$0.635120\pi$
$888$	0	0
$889$	2.47917	0.0831488
$890$	0	0
$891$	−7.32758	−0.245483
$892$	0	0
$893$	8.76031	0.293153
$894$	0	0
$895$	−77.7693	−2.59954
$896$	0	0
$897$	−7.23765	−0.241658
$898$	0	0
$899$	−5.57611	−0.185974
$900$	0	0
$901$	−43.4125	−1.44628
$902$	0	0
$903$	1.02121	0.0339837
$904$	0	0
$905$	69.1790	2.29959
$906$	0	0
$907$	26.2391	0.871254	0.435627	−	0.900127i	$-0.356527\pi$
$907$	26.2391	0.871254	0.435627	+	0.900127i	$0.356527\pi$
$908$	0	0
$909$	4.83226	0.160276
$910$	0	0
$911$	−33.9371	−1.12439	−0.562193	−	0.827006i	$-0.690042\pi$
$911$	−33.9371	−1.12439	−0.562193	+	0.827006i	$0.690042\pi$
$912$	0	0
$913$	−11.2926	−0.373730
$914$	0	0
$915$	−44.2862	−1.46406
$916$	0	0
$917$	3.17258	0.104768
$918$	0	0
$919$	6.62477	0.218531	0.109265	−	0.994013i	$-0.465150\pi$
$919$	6.62477	0.218531	0.109265	+	0.994013i	$0.465150\pi$
$920$	0	0
$921$	5.08751	0.167639
$922$	0	0
$923$	11.0185	0.362678
$924$	0	0
$925$	37.8765	1.24537
$926$	0	0
$927$	4.08680	0.134228
$928$	0	0
$929$	−48.5220	−1.59196	−0.795978	−	0.605326i	$-0.793043\pi$
$929$	−48.5220	−1.59196	−0.795978	+	0.605326i	$0.793043\pi$
$930$	0	0
$931$	−34.7920	−1.14026
$932$	0	0
$933$	−21.0632	−0.689577
$934$	0	0
$935$	51.8309	1.69505
$936$	0	0
$937$	2.56125	0.0836722	0.0418361	−	0.999124i	$-0.486679\pi$
$937$	2.56125	0.0836722	0.0418361	+	0.999124i	$0.486679\pi$
$938$	0	0
$939$	12.7984	0.417661
$940$	0	0
$941$	11.1766	0.364346	0.182173	−	0.983266i	$-0.441687\pi$
$941$	11.1766	0.364346	0.182173	+	0.983266i	$0.441687\pi$
$942$	0	0
$943$	7.72097	0.251429
$944$	0	0
$945$	4.91096	0.159753
$946$	0	0
$947$	25.6807	0.834509	0.417255	−	0.908790i	$-0.362992\pi$
$947$	25.6807	0.834509	0.417255	+	0.908790i	$0.362992\pi$
$948$	0	0
$949$	−21.3656	−0.693558
$950$	0	0
$951$	−19.5716	−0.634653
$952$	0	0
$953$	29.6835	0.961541	0.480771	−	0.876846i	$-0.340357\pi$
$953$	29.6835	0.961541	0.480771	+	0.876846i	$0.340357\pi$
$954$	0	0
$955$	41.7169	1.34993
$956$	0	0
$957$	−3.26735	−0.105619
$958$	0	0
$959$	2.06042	0.0665346
$960$	0	0
$961$	−12.7258	−0.410509
$962$	0	0
$963$	6.90474	0.222502
$964$	0	0
$965$	47.2625	1.52143
$966$	0	0
$967$	−4.93932	−0.158838	−0.0794189	−	0.996841i	$-0.525306\pi$
$967$	−4.93932	−0.158838	−0.0794189	+	0.996841i	$0.525306\pi$
$968$	0	0
$969$	−46.9907	−1.50956
$970$	0	0
$971$	−13.1712	−0.422684	−0.211342	−	0.977412i	$-0.567783\pi$
$971$	−13.1712	−0.422684	−0.211342	+	0.977412i	$0.567783\pi$
$972$	0	0
$973$	1.70137	0.0545433
$974$	0	0
$975$	−45.0357	−1.44230
$976$	0	0
$977$	−19.5422	−0.625210	−0.312605	−	0.949883i	$-0.601202\pi$
$977$	−19.5422	−0.625210	−0.312605	+	0.949883i	$0.601202\pi$
$978$	0	0
$979$	31.8749	1.01873
$980$	0	0
$981$	−5.66407	−0.180840
$982$	0	0
$983$	−53.8905	−1.71884	−0.859420	−	0.511270i	$-0.829175\pi$
$983$	−53.8905	−1.71884	−0.859420	+	0.511270i	$0.829175\pi$
$984$	0	0
$985$	−23.2558	−0.740992
$986$	0	0
$987$	−0.517215	−0.0164631
$988$	0	0
$989$	−5.88474	−0.187124
$990$	0	0
$991$	18.8625	0.599188	0.299594	−	0.954067i	$-0.403149\pi$
$991$	18.8625	0.599188	0.299594	+	0.954067i	$0.403149\pi$
$992$	0	0
$993$	−14.7565	−0.468284
$994$	0	0
$995$	−55.2557	−1.75172
$996$	0	0
$997$	−32.0914	−1.01634	−0.508172	−	0.861256i	$-0.669678\pi$
$997$	−32.0914	−1.01634	−0.508172	+	0.861256i	$0.669678\pi$
$998$	0	0
$999$	20.1359	0.637073

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.16	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.16	✓	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)

\(f(q)\)	\(=\)	\(q-1.33859 q^{3} +3.94917 q^{5} +0.220759 q^{7} -1.20816 q^{9} +O(q^{10})\)	\(q-1.33859 q^{3} +3.94917 q^{5} +0.220759 q^{7} -1.20816 q^{9} +1.87126 q^{11} +3.17517 q^{13} -5.28634 q^{15} +7.01370 q^{17} +5.00513 q^{19} -0.295507 q^{21} +1.70287 q^{23} +10.5960 q^{25} +5.63303 q^{27} +1.30440 q^{29} -4.27484 q^{31} -2.50486 q^{33} +0.871815 q^{35} +3.57462 q^{37} -4.25027 q^{39} +4.53410 q^{41} -3.45578 q^{43} -4.77125 q^{45} +1.75027 q^{47} -6.95127 q^{49} -9.38851 q^{51} -6.18967 q^{53} +7.38994 q^{55} -6.69984 q^{57} -10.8041 q^{59} +8.37748 q^{61} -0.266713 q^{63} +12.5393 q^{65} -3.87632 q^{67} -2.27945 q^{69} +3.47020 q^{71} -6.72897 q^{73} -14.1837 q^{75} +0.413098 q^{77} -6.08611 q^{79} -3.91584 q^{81} -6.03473 q^{83} +27.6983 q^{85} -1.74607 q^{87} +17.0339 q^{89} +0.700948 q^{91} +5.72227 q^{93} +19.7661 q^{95} +0.167792 q^{97} -2.26079 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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

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