LMFDB - Embedded newform 8024.2.a.x.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.02290 q^{3} +1.12353 q^{5} -1.44909 q^{7} +1.09211 q^{9} +O(q^{10})$	$q-2.02290 q^{3} +1.12353 q^{5} -1.44909 q^{7} +1.09211 q^{9} +0.323478 q^{11} -0.799377 q^{13} -2.27278 q^{15} -1.00000 q^{17} +4.33540 q^{19} +2.93136 q^{21} -2.09611 q^{23} -3.73769 q^{25} +3.85946 q^{27} -6.70077 q^{29} +5.47464 q^{31} -0.654363 q^{33} -1.62809 q^{35} +8.36966 q^{37} +1.61706 q^{39} +6.70873 q^{41} -10.7762 q^{43} +1.22702 q^{45} +11.0595 q^{47} -4.90014 q^{49} +2.02290 q^{51} -9.19339 q^{53} +0.363436 q^{55} -8.77006 q^{57} +1.00000 q^{59} -4.83808 q^{61} -1.58257 q^{63} -0.898121 q^{65} +10.0556 q^{67} +4.24022 q^{69} -10.5491 q^{71} -6.02857 q^{73} +7.56096 q^{75} -0.468748 q^{77} +9.70554 q^{79} -11.0836 q^{81} +13.3236 q^{83} -1.12353 q^{85} +13.5550 q^{87} +16.0752 q^{89} +1.15837 q^{91} -11.0746 q^{93} +4.87093 q^{95} +4.33283 q^{97} +0.353275 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * 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
$3$	−2.02290	−1.16792	−0.583960	−	0.811782i	$-0.698498\pi$
$3$	−2.02290	−1.16792	−0.583960	+	0.811782i	$0.698498\pi$
$4$	0	0
$5$	1.12353	0.502456	0.251228	−	0.967928i	$-0.419166\pi$
$5$	1.12353	0.502456	0.251228	+	0.967928i	$0.419166\pi$
$6$	0	0
$7$	−1.44909	−0.547704	−0.273852	−	0.961772i	$-0.588298\pi$
$7$	−1.44909	−0.547704	−0.273852	+	0.961772i	$0.588298\pi$
$8$	0	0
$9$	1.09211	0.364038
$10$	0	0
$11$	0.323478	0.0975323	0.0487662	−	0.998810i	$-0.484471\pi$
$11$	0.323478	0.0975323	0.0487662	+	0.998810i	$0.484471\pi$
$12$	0	0
$13$	−0.799377	−0.221707	−0.110854	−	0.993837i	$-0.535358\pi$
$13$	−0.799377	−0.221707	−0.110854	+	0.993837i	$0.535358\pi$
$14$	0	0
$15$	−2.27278	−0.586829
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	4.33540	0.994608	0.497304	−	0.867576i	$-0.334323\pi$
$19$	4.33540	0.994608	0.497304	+	0.867576i	$0.334323\pi$
$20$	0	0
$21$	2.93136	0.639674
$22$	0	0
$23$	−2.09611	−0.437070	−0.218535	−	0.975829i	$-0.570128\pi$
$23$	−2.09611	−0.437070	−0.218535	+	0.975829i	$0.570128\pi$
$24$	0	0
$25$	−3.73769	−0.747538
$26$	0	0
$27$	3.85946	0.742753
$28$	0	0
$29$	−6.70077	−1.24430	−0.622151	−	0.782898i	$-0.713741\pi$
$29$	−6.70077	−1.24430	−0.622151	+	0.782898i	$0.713741\pi$
$30$	0	0
$31$	5.47464	0.983275	0.491638	−	0.870800i	$-0.336398\pi$
$31$	5.47464	0.983275	0.491638	+	0.870800i	$0.336398\pi$
$32$	0	0
$33$	−0.654363	−0.113910
$34$	0	0
$35$	−1.62809	−0.275197
$36$	0	0
$37$	8.36966	1.37596	0.687982	−	0.725728i	$-0.258497\pi$
$37$	8.36966	1.37596	0.687982	+	0.725728i	$0.258497\pi$
$38$	0	0
$39$	1.61706	0.258936
$40$	0	0
$41$	6.70873	1.04773	0.523864	−	0.851802i	$-0.324490\pi$
$41$	6.70873	1.04773	0.523864	+	0.851802i	$0.324490\pi$
$42$	0	0
$43$	−10.7762	−1.64335	−0.821676	−	0.569955i	$-0.806961\pi$
$43$	−10.7762	−1.64335	−0.821676	+	0.569955i	$0.806961\pi$
$44$	0	0
$45$	1.22702	0.182913
$46$	0	0
$47$	11.0595	1.61319	0.806594	−	0.591105i	$-0.201308\pi$
$47$	11.0595	1.61319	0.806594	+	0.591105i	$0.201308\pi$
$48$	0	0
$49$	−4.90014	−0.700021
$50$	0	0
$51$	2.02290	0.283262
$52$	0	0
$53$	−9.19339	−1.26281	−0.631405	−	0.775453i	$-0.717521\pi$
$53$	−9.19339	−1.26281	−0.631405	+	0.775453i	$0.717521\pi$
$54$	0	0
$55$	0.363436	0.0490057
$56$	0	0
$57$	−8.77006	−1.16162
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−4.83808	−0.619453	−0.309727	−	0.950826i	$-0.600237\pi$
$61$	−4.83808	−0.619453	−0.309727	+	0.950826i	$0.600237\pi$
$62$	0	0
$63$	−1.58257	−0.199385
$64$	0	0
$65$	−0.898121	−0.111398
$66$	0	0
$67$	10.0556	1.22849	0.614244	−	0.789116i	$-0.289461\pi$
$67$	10.0556	1.22849	0.614244	+	0.789116i	$0.289461\pi$
$68$	0	0
$69$	4.24022	0.510463
$70$	0	0
$71$	−10.5491	−1.25194	−0.625972	−	0.779846i	$-0.715298\pi$
$71$	−10.5491	−1.25194	−0.625972	+	0.779846i	$0.715298\pi$
$72$	0	0
$73$	−6.02857	−0.705590	−0.352795	−	0.935701i	$-0.614769\pi$
$73$	−6.02857	−0.705590	−0.352795	+	0.935701i	$0.614769\pi$
$74$	0	0
$75$	7.56096	0.873064
$76$	0	0
$77$	−0.468748	−0.0534188
$78$	0	0
$79$	9.70554	1.09196	0.545979	−	0.837799i	$-0.316158\pi$
$79$	9.70554	1.09196	0.545979	+	0.837799i	$0.316158\pi$
$80$	0	0
$81$	−11.0836	−1.23151
$82$	0	0
$83$	13.3236	1.46246	0.731230	−	0.682131i	$-0.238947\pi$
$83$	13.3236	1.46246	0.731230	+	0.682131i	$0.238947\pi$
$84$	0	0
$85$	−1.12353	−0.121864
$86$	0	0
$87$	13.5550	1.45324
$88$	0	0
$89$	16.0752	1.70396	0.851982	−	0.523571i	$-0.175401\pi$
$89$	16.0752	1.70396	0.851982	+	0.523571i	$0.175401\pi$
$90$	0	0
$91$	1.15837	0.121430
$92$	0	0
$93$	−11.0746	−1.14839
$94$	0	0
$95$	4.87093	0.499747
$96$	0	0
$97$	4.33283	0.439932	0.219966	−	0.975508i	$-0.429405\pi$
$97$	4.33283	0.439932	0.219966	+	0.975508i	$0.429405\pi$
$98$	0	0
$99$	0.353275	0.0355054
$100$	0	0
$101$	1.43360	0.142648	0.0713240	−	0.997453i	$-0.477278\pi$
$101$	1.43360	0.142648	0.0713240	+	0.997453i	$0.477278\pi$
$102$	0	0
$103$	−5.10967	−0.503470	−0.251735	−	0.967796i	$-0.581001\pi$
$103$	−5.10967	−0.503470	−0.251735	+	0.967796i	$0.581001\pi$
$104$	0	0
$105$	3.29346	0.321408
$106$	0	0
$107$	−0.250194	−0.0241872	−0.0120936	−	0.999927i	$-0.503850\pi$
$107$	−0.250194	−0.0241872	−0.0120936	+	0.999927i	$0.503850\pi$
$108$	0	0
$109$	0.492894	0.0472107	0.0236054	−	0.999721i	$-0.492485\pi$
$109$	0.492894	0.0472107	0.0236054	+	0.999721i	$0.492485\pi$
$110$	0	0
$111$	−16.9310	−1.60702
$112$	0	0
$113$	−4.02351	−0.378500	−0.189250	−	0.981929i	$-0.560606\pi$
$113$	−4.02351	−0.378500	−0.189250	+	0.981929i	$0.560606\pi$
$114$	0	0
$115$	−2.35504	−0.219608
$116$	0	0
$117$	−0.873010	−0.0807098
$118$	0	0
$119$	1.44909	0.132838
$120$	0	0
$121$	−10.8954	−0.990487
$122$	0	0
$123$	−13.5711	−1.22366
$124$	0	0
$125$	−9.81702	−0.878061
$126$	0	0
$127$	6.09023	0.540420	0.270210	−	0.962801i	$-0.412907\pi$
$127$	6.09023	0.540420	0.270210	+	0.962801i	$0.412907\pi$
$128$	0	0
$129$	21.7991	1.91930
$130$	0	0
$131$	−16.9677	−1.48248	−0.741239	−	0.671241i	$-0.765761\pi$
$131$	−16.9677	−1.48248	−0.741239	+	0.671241i	$0.765761\pi$
$132$	0	0
$133$	−6.28237	−0.544751
$134$	0	0
$135$	4.33620	0.373201
$136$	0	0
$137$	7.00100	0.598136	0.299068	−	0.954232i	$-0.403324\pi$
$137$	7.00100	0.598136	0.299068	+	0.954232i	$0.403324\pi$
$138$	0	0
$139$	−16.2224	−1.37596	−0.687982	−	0.725728i	$-0.741503\pi$
$139$	−16.2224	−1.37596	−0.687982	+	0.725728i	$0.741503\pi$
$140$	0	0
$141$	−22.3722	−1.88408
$142$	0	0
$143$	−0.258581	−0.0216236
$144$	0	0
$145$	−7.52849	−0.625207
$146$	0	0
$147$	9.91249	0.817568
$148$	0	0
$149$	−11.4679	−0.939491	−0.469745	−	0.882802i	$-0.655654\pi$
$149$	−11.4679	−0.939491	−0.469745	+	0.882802i	$0.655654\pi$
$150$	0	0
$151$	−12.3141	−1.00210	−0.501052	−	0.865417i	$-0.667053\pi$
$151$	−12.3141	−1.00210	−0.501052	+	0.865417i	$0.667053\pi$
$152$	0	0
$153$	−1.09211	−0.0882921
$154$	0	0
$155$	6.15091	0.494053
$156$	0	0
$157$	4.56128	0.364030	0.182015	−	0.983296i	$-0.441738\pi$
$157$	4.56128	0.364030	0.182015	+	0.983296i	$0.441738\pi$
$158$	0	0
$159$	18.5973	1.47486
$160$	0	0
$161$	3.03745	0.239385
$162$	0	0
$163$	15.0140	1.17599	0.587994	−	0.808866i	$-0.299918\pi$
$163$	15.0140	1.17599	0.587994	+	0.808866i	$0.299918\pi$
$164$	0	0
$165$	−0.735194	−0.0572348
$166$	0	0
$167$	6.92953	0.536223	0.268111	−	0.963388i	$-0.413600\pi$
$167$	6.92953	0.536223	0.268111	+	0.963388i	$0.413600\pi$
$168$	0	0
$169$	−12.3610	−0.950846
$170$	0	0
$171$	4.73474	0.362075
$172$	0	0
$173$	−10.4434	−0.794001	−0.397000	−	0.917818i	$-0.629949\pi$
$173$	−10.4434	−0.794001	−0.397000	+	0.917818i	$0.629949\pi$
$174$	0	0
$175$	5.41624	0.409429
$176$	0	0
$177$	−2.02290	−0.152050
$178$	0	0
$179$	16.6116	1.24161	0.620806	−	0.783965i	$-0.286806\pi$
$179$	16.6116	1.24161	0.620806	+	0.783965i	$0.286806\pi$
$180$	0	0
$181$	1.91658	0.142458	0.0712291	−	0.997460i	$-0.477308\pi$
$181$	1.91658	0.142458	0.0712291	+	0.997460i	$0.477308\pi$
$182$	0	0
$183$	9.78694	0.723472
$184$	0	0
$185$	9.40353	0.691361
$186$	0	0
$187$	−0.323478	−0.0236551
$188$	0	0
$189$	−5.59270	−0.406809
$190$	0	0
$191$	8.91934	0.645380	0.322690	−	0.946505i	$-0.395413\pi$
$191$	8.91934	0.645380	0.322690	+	0.946505i	$0.395413\pi$
$192$	0	0
$193$	−19.9791	−1.43812	−0.719062	−	0.694946i	$-0.755428\pi$
$193$	−19.9791	−1.43812	−0.719062	+	0.694946i	$0.755428\pi$
$194$	0	0
$195$	1.81681	0.130104
$196$	0	0
$197$	−6.73635	−0.479945	−0.239973	−	0.970780i	$-0.577138\pi$
$197$	−6.73635	−0.479945	−0.239973	+	0.970780i	$0.577138\pi$
$198$	0	0
$199$	7.94595	0.563273	0.281637	−	0.959521i	$-0.409123\pi$
$199$	7.94595	0.563273	0.281637	+	0.959521i	$0.409123\pi$
$200$	0	0
$201$	−20.3415	−1.43478
$202$	0	0
$203$	9.71000	0.681509
$204$	0	0
$205$	7.53744	0.526438
$206$	0	0
$207$	−2.28919	−0.159110
$208$	0	0
$209$	1.40241	0.0970065
$210$	0	0
$211$	−6.42028	−0.441990	−0.220995	−	0.975275i	$-0.570930\pi$
$211$	−6.42028	−0.441990	−0.220995	+	0.975275i	$0.570930\pi$
$212$	0	0
$213$	21.3397	1.46217
$214$	0	0
$215$	−12.1073	−0.825712
$216$	0	0
$217$	−7.93324	−0.538544
$218$	0	0
$219$	12.1952	0.824073
$220$	0	0
$221$	0.799377	0.0537719
$222$	0	0
$223$	18.1621	1.21623	0.608113	−	0.793851i	$-0.291927\pi$
$223$	18.1621	1.21623	0.608113	+	0.793851i	$0.291927\pi$
$224$	0	0
$225$	−4.08198	−0.272132
$226$	0	0
$227$	16.4847	1.09413	0.547065	−	0.837090i	$-0.315745\pi$
$227$	16.4847	1.09413	0.547065	+	0.837090i	$0.315745\pi$
$228$	0	0
$229$	21.8282	1.44245	0.721225	−	0.692701i	$-0.243579\pi$
$229$	21.8282	1.44245	0.721225	+	0.692701i	$0.243579\pi$
$230$	0	0
$231$	0.948230	0.0623889
$232$	0	0
$233$	−11.9244	−0.781191	−0.390596	−	0.920562i	$-0.627731\pi$
$233$	−11.9244	−0.781191	−0.390596	+	0.920562i	$0.627731\pi$
$234$	0	0
$235$	12.4256	0.810557
$236$	0	0
$237$	−19.6333	−1.27532
$238$	0	0
$239$	−5.43866	−0.351798	−0.175899	−	0.984408i	$-0.556283\pi$
$239$	−5.43866	−0.351798	−0.175899	+	0.984408i	$0.556283\pi$
$240$	0	0
$241$	−2.91172	−0.187560	−0.0937802	−	0.995593i	$-0.529895\pi$
$241$	−2.91172	−0.187560	−0.0937802	+	0.995593i	$0.529895\pi$
$242$	0	0
$243$	10.8427	0.695557
$244$	0	0
$245$	−5.50544	−0.351730
$246$	0	0
$247$	−3.46562	−0.220512
$248$	0	0
$249$	−26.9524	−1.70804
$250$	0	0
$251$	−25.3273	−1.59864	−0.799322	−	0.600903i	$-0.794808\pi$
$251$	−25.3273	−1.59864	−0.799322	+	0.600903i	$0.794808\pi$
$252$	0	0
$253$	−0.678047	−0.0426284
$254$	0	0
$255$	2.27278	0.142327
$256$	0	0
$257$	1.58786	0.0990481	0.0495240	−	0.998773i	$-0.484230\pi$
$257$	1.58786	0.0990481	0.0495240	+	0.998773i	$0.484230\pi$
$258$	0	0
$259$	−12.1284	−0.753620
$260$	0	0
$261$	−7.31800	−0.452973
$262$	0	0
$263$	2.65253	0.163562	0.0817810	−	0.996650i	$-0.473939\pi$
$263$	2.65253	0.163562	0.0817810	+	0.996650i	$0.473939\pi$
$264$	0	0
$265$	−10.3290	−0.634507
$266$	0	0
$267$	−32.5184	−1.99009
$268$	0	0
$269$	6.39842	0.390118	0.195059	−	0.980791i	$-0.437510\pi$
$269$	6.39842	0.390118	0.195059	+	0.980791i	$0.437510\pi$
$270$	0	0
$271$	−24.6124	−1.49510	−0.747549	−	0.664207i	$-0.768769\pi$
$271$	−24.6124	−1.49510	−0.747549	+	0.664207i	$0.768769\pi$
$272$	0	0
$273$	−2.34326	−0.141820
$274$	0	0
$275$	−1.20906	−0.0729091
$276$	0	0
$277$	3.77521	0.226831	0.113415	−	0.993548i	$-0.463821\pi$
$277$	3.77521	0.226831	0.113415	+	0.993548i	$0.463821\pi$
$278$	0	0
$279$	5.97893	0.357949
$280$	0	0
$281$	−19.0800	−1.13822	−0.569109	−	0.822262i	$-0.692712\pi$
$281$	−19.0800	−1.13822	−0.569109	+	0.822262i	$0.692712\pi$
$282$	0	0
$283$	4.61213	0.274163	0.137081	−	0.990560i	$-0.456228\pi$
$283$	4.61213	0.274163	0.137081	+	0.990560i	$0.456228\pi$
$284$	0	0
$285$	−9.85340	−0.583665
$286$	0	0
$287$	−9.72155	−0.573845
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−8.76486	−0.513805
$292$	0	0
$293$	11.6116	0.678359	0.339179	−	0.940722i	$-0.389851\pi$
$293$	11.6116	0.678359	0.339179	+	0.940722i	$0.389851\pi$
$294$	0	0
$295$	1.12353	0.0654142
$296$	0	0
$297$	1.24845	0.0724425
$298$	0	0
$299$	1.67558	0.0969015
$300$	0	0
$301$	15.6156	0.900070
$302$	0	0
$303$	−2.90002	−0.166602
$304$	0	0
$305$	−5.43571	−0.311248
$306$	0	0
$307$	−9.00375	−0.513871	−0.256936	−	0.966429i	$-0.582713\pi$
$307$	−9.00375	−0.513871	−0.256936	+	0.966429i	$0.582713\pi$
$308$	0	0
$309$	10.3363	0.588013
$310$	0	0
$311$	17.5315	0.994118	0.497059	−	0.867717i	$-0.334413\pi$
$311$	17.5315	0.994118	0.497059	+	0.867717i	$0.334413\pi$
$312$	0	0
$313$	24.3946	1.37886	0.689431	−	0.724351i	$-0.257861\pi$
$313$	24.3946	1.37886	0.689431	+	0.724351i	$0.257861\pi$
$314$	0	0
$315$	−1.77806	−0.100182
$316$	0	0
$317$	−1.87990	−0.105586	−0.0527928	−	0.998605i	$-0.516812\pi$
$317$	−1.87990	−0.105586	−0.0527928	+	0.998605i	$0.516812\pi$
$318$	0	0
$319$	−2.16755	−0.121360
$320$	0	0
$321$	0.506117	0.0282487
$322$	0	0
$323$	−4.33540	−0.241228
$324$	0	0
$325$	2.98782	0.165734
$326$	0	0
$327$	−0.997075	−0.0551384
$328$	0	0
$329$	−16.0261	−0.883550
$330$	0	0
$331$	−17.5083	−0.962342	−0.481171	−	0.876627i	$-0.659788\pi$
$331$	−17.5083	−0.962342	−0.481171	+	0.876627i	$0.659788\pi$
$332$	0	0
$333$	9.14062	0.500903
$334$	0	0
$335$	11.2977	0.617262
$336$	0	0
$337$	−24.5306	−1.33627	−0.668133	−	0.744042i	$-0.732906\pi$
$337$	−24.5306	−1.33627	−0.668133	+	0.744042i	$0.732906\pi$
$338$	0	0
$339$	8.13915	0.442058
$340$	0	0
$341$	1.77093	0.0959011
$342$	0	0
$343$	17.2444	0.931108
$344$	0	0
$345$	4.76400	0.256485
$346$	0	0
$347$	−21.7982	−1.17019	−0.585095	−	0.810965i	$-0.698943\pi$
$347$	−21.7982	−1.17019	−0.585095	+	0.810965i	$0.698943\pi$
$348$	0	0
$349$	−15.4685	−0.828011	−0.414005	−	0.910274i	$-0.635871\pi$
$349$	−15.4685	−0.828011	−0.414005	+	0.910274i	$0.635871\pi$
$350$	0	0
$351$	−3.08516	−0.164674
$352$	0	0
$353$	3.10249	0.165129	0.0825643	−	0.996586i	$-0.473689\pi$
$353$	3.10249	0.165129	0.0825643	+	0.996586i	$0.473689\pi$
$354$	0	0
$355$	−11.8522	−0.629047
$356$	0	0
$357$	−2.93136	−0.155144
$358$	0	0
$359$	−35.9381	−1.89674	−0.948369	−	0.317169i	$-0.897268\pi$
$359$	−35.9381	−1.89674	−0.948369	+	0.317169i	$0.897268\pi$
$360$	0	0
$361$	−0.204328	−0.0107541
$362$	0	0
$363$	22.0402	1.15681
$364$	0	0
$365$	−6.77325	−0.354528
$366$	0	0
$367$	−33.2664	−1.73649	−0.868245	−	0.496136i	$-0.834752\pi$
$367$	−33.2664	−1.73649	−0.868245	+	0.496136i	$0.834752\pi$
$368$	0	0
$369$	7.32670	0.381413
$370$	0	0
$371$	13.3220	0.691646
$372$	0	0
$373$	−25.2963	−1.30979	−0.654897	−	0.755718i	$-0.727288\pi$
$373$	−25.2963	−1.30979	−0.654897	+	0.755718i	$0.727288\pi$
$374$	0	0
$375$	19.8588	1.02551
$376$	0	0
$377$	5.35644	0.275871
$378$	0	0
$379$	16.7858	0.862230	0.431115	−	0.902297i	$-0.358120\pi$
$379$	16.7858	0.862230	0.431115	+	0.902297i	$0.358120\pi$
$380$	0	0
$381$	−12.3199	−0.631168
$382$	0	0
$383$	3.15808	0.161370	0.0806852	−	0.996740i	$-0.474289\pi$
$383$	3.15808	0.161370	0.0806852	+	0.996740i	$0.474289\pi$
$384$	0	0
$385$	−0.526651	−0.0268406
$386$	0	0
$387$	−11.7688	−0.598242
$388$	0	0
$389$	−33.4517	−1.69607	−0.848034	−	0.529942i	$-0.822214\pi$
$389$	−33.4517	−1.69607	−0.848034	+	0.529942i	$0.822214\pi$
$390$	0	0
$391$	2.09611	0.106005
$392$	0	0
$393$	34.3240	1.73142
$394$	0	0
$395$	10.9044	0.548662
$396$	0	0
$397$	15.5802	0.781948	0.390974	−	0.920402i	$-0.372138\pi$
$397$	15.5802	0.781948	0.390974	+	0.920402i	$0.372138\pi$
$398$	0	0
$399$	12.7086	0.636225
$400$	0	0
$401$	11.1801	0.558309	0.279154	−	0.960246i	$-0.409946\pi$
$401$	11.1801	0.558309	0.279154	+	0.960246i	$0.409946\pi$
$402$	0	0
$403$	−4.37630	−0.217999
$404$	0	0
$405$	−12.4527	−0.618782
$406$	0	0
$407$	2.70740	0.134201
$408$	0	0
$409$	−10.2588	−0.507264	−0.253632	−	0.967301i	$-0.581625\pi$
$409$	−10.2588	−0.507264	−0.253632	+	0.967301i	$0.581625\pi$
$410$	0	0
$411$	−14.1623	−0.698575
$412$	0	0
$413$	−1.44909	−0.0713050
$414$	0	0
$415$	14.9695	0.734822
$416$	0	0
$417$	32.8162	1.60702
$418$	0	0
$419$	5.94371	0.290369	0.145185	−	0.989405i	$-0.453622\pi$
$419$	5.94371	0.290369	0.145185	+	0.989405i	$0.453622\pi$
$420$	0	0
$421$	−3.33937	−0.162751	−0.0813755	−	0.996684i	$-0.525931\pi$
$421$	−3.33937	−0.162751	−0.0813755	+	0.996684i	$0.525931\pi$
$422$	0	0
$423$	12.0782	0.587261
$424$	0	0
$425$	3.73769	0.181305
$426$	0	0
$427$	7.01081	0.339277
$428$	0	0
$429$	0.523082	0.0252547
$430$	0	0
$431$	−6.93142	−0.333875	−0.166937	−	0.985968i	$-0.553388\pi$
$431$	−6.93142	−0.333875	−0.166937	+	0.985968i	$0.553388\pi$
$432$	0	0
$433$	−15.1068	−0.725987	−0.362994	−	0.931792i	$-0.618245\pi$
$433$	−15.1068	−0.725987	−0.362994	+	0.931792i	$0.618245\pi$
$434$	0	0
$435$	15.2294	0.730192
$436$	0	0
$437$	−9.08748	−0.434713
$438$	0	0
$439$	−37.6029	−1.79469	−0.897344	−	0.441332i	$-0.854506\pi$
$439$	−37.6029	−1.79469	−0.897344	+	0.441332i	$0.854506\pi$
$440$	0	0
$441$	−5.35151	−0.254834
$442$	0	0
$443$	−10.6913	−0.507959	−0.253979	−	0.967210i	$-0.581740\pi$
$443$	−10.6913	−0.507959	−0.253979	+	0.967210i	$0.581740\pi$
$444$	0	0
$445$	18.0609	0.856167
$446$	0	0
$447$	23.1985	1.09725
$448$	0	0
$449$	−25.5274	−1.20471	−0.602356	−	0.798228i	$-0.705771\pi$
$449$	−25.5274	−1.20471	−0.602356	+	0.798228i	$0.705771\pi$
$450$	0	0
$451$	2.17013	0.102187
$452$	0	0
$453$	24.9101	1.17038
$454$	0	0
$455$	1.30146	0.0610132
$456$	0	0
$457$	−14.0473	−0.657107	−0.328554	−	0.944485i	$-0.606561\pi$
$457$	−14.0473	−0.657107	−0.328554	+	0.944485i	$0.606561\pi$
$458$	0	0
$459$	−3.85946	−0.180144
$460$	0	0
$461$	−14.9826	−0.697809	−0.348905	−	0.937158i	$-0.613446\pi$
$461$	−14.9826	−0.697809	−0.348905	+	0.937158i	$0.613446\pi$
$462$	0	0
$463$	30.9769	1.43962	0.719808	−	0.694173i	$-0.244230\pi$
$463$	30.9769	1.43962	0.719808	+	0.694173i	$0.244230\pi$
$464$	0	0
$465$	−12.4427	−0.577014
$466$	0	0
$467$	−29.9499	−1.38592	−0.692958	−	0.720978i	$-0.743693\pi$
$467$	−29.9499	−1.38592	−0.692958	+	0.720978i	$0.743693\pi$
$468$	0	0
$469$	−14.5715	−0.672848
$470$	0	0
$471$	−9.22700	−0.425158
$472$	0	0
$473$	−3.48586	−0.160280
$474$	0	0
$475$	−16.2044	−0.743507
$476$	0	0
$477$	−10.0402	−0.459710
$478$	0	0
$479$	9.40212	0.429594	0.214797	−	0.976659i	$-0.431091\pi$
$479$	9.40212	0.429594	0.214797	+	0.976659i	$0.431091\pi$
$480$	0	0
$481$	−6.69051	−0.305061
$482$	0	0
$483$	−6.14445	−0.279582
$484$	0	0
$485$	4.86804	0.221047
$486$	0	0
$487$	−6.98982	−0.316739	−0.158370	−	0.987380i	$-0.550624\pi$
$487$	−6.98982	−0.316739	−0.158370	+	0.987380i	$0.550624\pi$
$488$	0	0
$489$	−30.3718	−1.37346
$490$	0	0
$491$	−31.5892	−1.42560	−0.712801	−	0.701367i	$-0.752574\pi$
$491$	−31.5892	−1.42560	−0.712801	+	0.701367i	$0.752574\pi$
$492$	0	0
$493$	6.70077	0.301787
$494$	0	0
$495$	0.396913	0.0178399
$496$	0	0
$497$	15.2865	0.685694
$498$	0	0
$499$	−27.1060	−1.21343	−0.606715	−	0.794920i	$-0.707513\pi$
$499$	−27.1060	−1.21343	−0.606715	+	0.794920i	$0.707513\pi$
$500$	0	0
$501$	−14.0177	−0.626265
$502$	0	0
$503$	−25.1335	−1.12065	−0.560324	−	0.828273i	$-0.689323\pi$
$503$	−25.1335	−1.12065	−0.560324	+	0.828273i	$0.689323\pi$
$504$	0	0
$505$	1.61068	0.0716744
$506$	0	0
$507$	25.0050	1.11051
$508$	0	0
$509$	20.8697	0.925035	0.462518	−	0.886610i	$-0.346946\pi$
$509$	20.8697	0.925035	0.462518	+	0.886610i	$0.346946\pi$
$510$	0	0
$511$	8.73592	0.386455
$512$	0	0
$513$	16.7323	0.738749
$514$	0	0
$515$	−5.74084	−0.252972
$516$	0	0
$517$	3.57749	0.157338
$518$	0	0
$519$	21.1260	0.927330
$520$	0	0
$521$	21.9277	0.960668	0.480334	−	0.877086i	$-0.340515\pi$
$521$	21.9277	0.960668	0.480334	+	0.877086i	$0.340515\pi$
$522$	0	0
$523$	−23.1557	−1.01253	−0.506265	−	0.862378i	$-0.668974\pi$
$523$	−23.1557	−1.01253	−0.506265	+	0.862378i	$0.668974\pi$
$524$	0	0
$525$	−10.9565	−0.478181
$526$	0	0
$527$	−5.47464	−0.238479
$528$	0	0
$529$	−18.6063	−0.808970
$530$	0	0
$531$	1.09211	0.0473937
$532$	0	0
$533$	−5.36281	−0.232289
$534$	0	0
$535$	−0.281100	−0.0121530
$536$	0	0
$537$	−33.6036	−1.45010
$538$	0	0
$539$	−1.58509	−0.0682746
$540$	0	0
$541$	−10.4619	−0.449791	−0.224896	−	0.974383i	$-0.572204\pi$
$541$	−10.4619	−0.449791	−0.224896	+	0.974383i	$0.572204\pi$
$542$	0	0
$543$	−3.87704	−0.166380
$544$	0	0
$545$	0.553780	0.0237213
$546$	0	0
$547$	38.1510	1.63122	0.815610	−	0.578603i	$-0.196402\pi$
$547$	38.1510	1.63122	0.815610	+	0.578603i	$0.196402\pi$
$548$	0	0
$549$	−5.28373	−0.225504
$550$	0	0
$551$	−29.0505	−1.23759
$552$	0	0
$553$	−14.0642	−0.598070
$554$	0	0
$555$	−19.0224	−0.807455
$556$	0	0
$557$	0.978688	0.0414683	0.0207342	−	0.999785i	$-0.493400\pi$
$557$	0.978688	0.0414683	0.0207342	+	0.999785i	$0.493400\pi$
$558$	0	0
$559$	8.61423	0.364343
$560$	0	0
$561$	0.654363	0.0276272
$562$	0	0
$563$	38.4585	1.62083	0.810416	−	0.585855i	$-0.199241\pi$
$563$	38.4585	1.62083	0.810416	+	0.585855i	$0.199241\pi$
$564$	0	0
$565$	−4.52052	−0.190180
$566$	0	0
$567$	16.0612	0.674505
$568$	0	0
$569$	−13.4028	−0.561876	−0.280938	−	0.959726i	$-0.590645\pi$
$569$	−13.4028	−0.561876	−0.280938	+	0.959726i	$0.590645\pi$
$570$	0	0
$571$	4.39655	0.183990	0.0919949	−	0.995759i	$-0.470676\pi$
$571$	4.39655	0.183990	0.0919949	+	0.995759i	$0.470676\pi$
$572$	0	0
$573$	−18.0429	−0.753753
$574$	0	0
$575$	7.83462	0.326726
$576$	0	0
$577$	1.41694	0.0589879	0.0294940	−	0.999565i	$-0.490610\pi$
$577$	1.41694	0.0589879	0.0294940	+	0.999565i	$0.490610\pi$
$578$	0	0
$579$	40.4156	1.67961
$580$	0	0
$581$	−19.3071	−0.800995
$582$	0	0
$583$	−2.97386	−0.123165
$584$	0	0
$585$	−0.980849	−0.0405531
$586$	0	0
$587$	−11.4632	−0.473135	−0.236568	−	0.971615i	$-0.576022\pi$
$587$	−11.4632	−0.473135	−0.236568	+	0.971615i	$0.576022\pi$
$588$	0	0
$589$	23.7348	0.977974
$590$	0	0
$591$	13.6269	0.560538
$592$	0	0
$593$	−19.4375	−0.798203	−0.399101	−	0.916907i	$-0.630678\pi$
$593$	−19.4375	−0.798203	−0.399101	+	0.916907i	$0.630678\pi$
$594$	0	0
$595$	1.62809	0.0667451
$596$	0	0
$597$	−16.0738	−0.657858
$598$	0	0
$599$	−27.8302	−1.13711	−0.568555	−	0.822645i	$-0.692497\pi$
$599$	−27.8302	−1.13711	−0.568555	+	0.822645i	$0.692497\pi$
$600$	0	0
$601$	13.9927	0.570776	0.285388	−	0.958412i	$-0.407878\pi$
$601$	13.9927	0.570776	0.285388	+	0.958412i	$0.407878\pi$
$602$	0	0
$603$	10.9819	0.447216
$604$	0	0
$605$	−12.2412	−0.497677
$606$	0	0
$607$	−30.1213	−1.22259	−0.611293	−	0.791404i	$-0.709350\pi$
$607$	−30.1213	−1.22259	−0.611293	+	0.791404i	$0.709350\pi$
$608$	0	0
$609$	−19.6423	−0.795948
$610$	0	0
$611$	−8.84068	−0.357656
$612$	0	0
$613$	−14.9510	−0.603867	−0.301933	−	0.953329i	$-0.597632\pi$
$613$	−14.9510	−0.603867	−0.301933	+	0.953329i	$0.597632\pi$
$614$	0	0
$615$	−15.2475	−0.614837
$616$	0	0
$617$	−40.0120	−1.61082	−0.805411	−	0.592717i	$-0.798055\pi$
$617$	−40.0120	−1.61082	−0.805411	+	0.592717i	$0.798055\pi$
$618$	0	0
$619$	35.9638	1.44551	0.722753	−	0.691106i	$-0.242876\pi$
$619$	35.9638	1.44551	0.722753	+	0.691106i	$0.242876\pi$
$620$	0	0
$621$	−8.08986	−0.324635
$622$	0	0
$623$	−23.2943	−0.933268
$624$	0	0
$625$	7.65876	0.306350
$626$	0	0
$627$	−2.83692	−0.113296
$628$	0	0
$629$	−8.36966	−0.333720
$630$	0	0
$631$	−48.2055	−1.91903	−0.959516	−	0.281655i	$-0.909117\pi$
$631$	−48.2055	−1.91903	−0.959516	+	0.281655i	$0.909117\pi$
$632$	0	0
$633$	12.9876	0.516209
$634$	0	0
$635$	6.84253	0.271537
$636$	0	0
$637$	3.91706	0.155200
$638$	0	0
$639$	−11.5208	−0.455755
$640$	0	0
$641$	−19.8458	−0.783860	−0.391930	−	0.919995i	$-0.628192\pi$
$641$	−19.8458	−0.783860	−0.391930	+	0.919995i	$0.628192\pi$
$642$	0	0
$643$	−25.4842	−1.00500	−0.502499	−	0.864578i	$-0.667586\pi$
$643$	−25.4842	−1.00500	−0.502499	+	0.864578i	$0.667586\pi$
$644$	0	0
$645$	24.4919	0.964366
$646$	0	0
$647$	3.86819	0.152074	0.0760372	−	0.997105i	$-0.475773\pi$
$647$	3.86819	0.152074	0.0760372	+	0.997105i	$0.475773\pi$
$648$	0	0
$649$	0.323478	0.0126976
$650$	0	0
$651$	16.0481	0.628976
$652$	0	0
$653$	−7.74732	−0.303176	−0.151588	−	0.988444i	$-0.548439\pi$
$653$	−7.74732	−0.303176	−0.151588	+	0.988444i	$0.548439\pi$
$654$	0	0
$655$	−19.0637	−0.744880
$656$	0	0
$657$	−6.58388	−0.256861
$658$	0	0
$659$	35.4579	1.38125	0.690623	−	0.723215i	$-0.257337\pi$
$659$	35.4579	1.38125	0.690623	+	0.723215i	$0.257337\pi$
$660$	0	0
$661$	−1.92073	−0.0747078	−0.0373539	−	0.999302i	$-0.511893\pi$
$661$	−1.92073	−0.0747078	−0.0373539	+	0.999302i	$0.511893\pi$
$662$	0	0
$663$	−1.61706	−0.0628013
$664$	0	0
$665$	−7.05841	−0.273713
$666$	0	0
$667$	14.0456	0.543846
$668$	0	0
$669$	−36.7401	−1.42045
$670$	0	0
$671$	−1.56501	−0.0604167
$672$	0	0
$673$	−8.54435	−0.329361	−0.164680	−	0.986347i	$-0.552659\pi$
$673$	−8.54435	−0.329361	−0.164680	+	0.986347i	$0.552659\pi$
$674$	0	0
$675$	−14.4255	−0.555236
$676$	0	0
$677$	26.2227	1.00782	0.503911	−	0.863756i	$-0.331894\pi$
$677$	26.2227	1.00782	0.503911	+	0.863756i	$0.331894\pi$
$678$	0	0
$679$	−6.27865	−0.240952
$680$	0	0
$681$	−33.3469	−1.27786
$682$	0	0
$683$	5.31640	0.203426	0.101713	−	0.994814i	$-0.467568\pi$
$683$	5.31640	0.203426	0.101713	+	0.994814i	$0.467568\pi$
$684$	0	0
$685$	7.86581	0.300537
$686$	0	0
$687$	−44.1563	−1.68467
$688$	0	0
$689$	7.34898	0.279974
$690$	0	0
$691$	46.9291	1.78527	0.892634	−	0.450782i	$-0.148855\pi$
$691$	46.9291	1.78527	0.892634	+	0.450782i	$0.148855\pi$
$692$	0	0
$693$	−0.511926	−0.0194465
$694$	0	0
$695$	−18.2263	−0.691362
$696$	0	0
$697$	−6.70873	−0.254111
$698$	0	0
$699$	24.1218	0.912369
$700$	0	0
$701$	−37.2529	−1.40702	−0.703511	−	0.710684i	$-0.748386\pi$
$701$	−37.2529	−1.40702	−0.703511	+	0.710684i	$0.748386\pi$
$702$	0	0
$703$	36.2858	1.36854
$704$	0	0
$705$	−25.1357	−0.946666
$706$	0	0
$707$	−2.07741	−0.0781289
$708$	0	0
$709$	12.6852	0.476404	0.238202	−	0.971216i	$-0.423442\pi$
$709$	12.6852	0.476404	0.238202	+	0.971216i	$0.423442\pi$
$710$	0	0
$711$	10.5996	0.397514
$712$	0	0
$713$	−11.4755	−0.429760
$714$	0	0
$715$	−0.290522	−0.0108649
$716$	0	0
$717$	11.0019	0.410872
$718$	0	0
$719$	−7.96365	−0.296994	−0.148497	−	0.988913i	$-0.547444\pi$
$719$	−7.96365	−0.296994	−0.148497	+	0.988913i	$0.547444\pi$
$720$	0	0
$721$	7.40435	0.275753
$722$	0	0
$723$	5.89011	0.219056
$724$	0	0
$725$	25.0454	0.930162
$726$	0	0
$727$	−37.3819	−1.38642	−0.693208	−	0.720737i	$-0.743803\pi$
$727$	−37.3819	−1.38642	−0.693208	+	0.720737i	$0.743803\pi$
$728$	0	0
$729$	11.3173	0.419159
$730$	0	0
$731$	10.7762	0.398571
$732$	0	0
$733$	14.9313	0.551498	0.275749	−	0.961230i	$-0.411074\pi$
$733$	14.9313	0.551498	0.275749	+	0.961230i	$0.411074\pi$
$734$	0	0
$735$	11.1369	0.410792
$736$	0	0
$737$	3.25277	0.119817
$738$	0	0
$739$	−3.92105	−0.144238	−0.0721190	−	0.997396i	$-0.522976\pi$
$739$	−3.92105	−0.144238	−0.0721190	+	0.997396i	$0.522976\pi$
$740$	0	0
$741$	7.01058	0.257540
$742$	0	0
$743$	−28.1721	−1.03354	−0.516768	−	0.856126i	$-0.672865\pi$
$743$	−28.1721	−1.03354	−0.516768	+	0.856126i	$0.672865\pi$
$744$	0	0
$745$	−12.8845	−0.472053
$746$	0	0
$747$	14.5509	0.532391
$748$	0	0
$749$	0.362553	0.0132474
$750$	0	0
$751$	35.1620	1.28308	0.641541	−	0.767089i	$-0.278296\pi$
$751$	35.1620	1.28308	0.641541	+	0.767089i	$0.278296\pi$
$752$	0	0
$753$	51.2345	1.86709
$754$	0	0
$755$	−13.8352	−0.503513
$756$	0	0
$757$	3.85719	0.140192	0.0700959	−	0.997540i	$-0.477669\pi$
$757$	3.85719	0.140192	0.0700959	+	0.997540i	$0.477669\pi$
$758$	0	0
$759$	1.37162	0.0497866
$760$	0	0
$761$	0.632247	0.0229189	0.0114595	−	0.999934i	$-0.496352\pi$
$761$	0.632247	0.0229189	0.0114595	+	0.999934i	$0.496352\pi$
$762$	0	0
$763$	−0.714247	−0.0258575
$764$	0	0
$765$	−1.22702	−0.0443629
$766$	0	0
$767$	−0.799377	−0.0288638
$768$	0	0
$769$	−7.65481	−0.276039	−0.138020	−	0.990429i	$-0.544074\pi$
$769$	−7.65481	−0.276039	−0.138020	+	0.990429i	$0.544074\pi$
$770$	0	0
$771$	−3.21208	−0.115680
$772$	0	0
$773$	−7.22520	−0.259872	−0.129936	−	0.991522i	$-0.541477\pi$
$773$	−7.22520	−0.259872	−0.129936	+	0.991522i	$0.541477\pi$
$774$	0	0
$775$	−20.4625	−0.735035
$776$	0	0
$777$	24.5345	0.880169
$778$	0	0
$779$	29.0850	1.04208
$780$	0	0
$781$	−3.41239	−0.122105
$782$	0	0
$783$	−25.8613	−0.924209
$784$	0	0
$785$	5.12472	0.182909
$786$	0	0
$787$	25.7359	0.917387	0.458693	−	0.888595i	$-0.348318\pi$
$787$	25.7359	0.917387	0.458693	+	0.888595i	$0.348318\pi$
$788$	0	0
$789$	−5.36579	−0.191027
$790$	0	0
$791$	5.83042	0.207306
$792$	0	0
$793$	3.86745	0.137337
$794$	0	0
$795$	20.8945	0.741053
$796$	0	0
$797$	45.7276	1.61975	0.809877	−	0.586600i	$-0.199534\pi$
$797$	45.7276	1.61975	0.809877	+	0.586600i	$0.199534\pi$
$798$	0	0
$799$	−11.0595	−0.391256
$800$	0	0
$801$	17.5559	0.620307
$802$	0	0
$803$	−1.95011	−0.0688179
$804$	0	0
$805$	3.41266	0.120280
$806$	0	0
$807$	−12.9433	−0.455627
$808$	0	0
$809$	−41.6106	−1.46295	−0.731476	−	0.681867i	$-0.761168\pi$
$809$	−41.6106	−1.46295	−0.731476	+	0.681867i	$0.761168\pi$
$810$	0	0
$811$	−12.3896	−0.435057	−0.217528	−	0.976054i	$-0.569799\pi$
$811$	−12.3896	−0.435057	−0.217528	+	0.976054i	$0.569799\pi$
$812$	0	0
$813$	49.7884	1.74615
$814$	0	0
$815$	16.8686	0.590882
$816$	0	0
$817$	−46.7190	−1.63449
$818$	0	0
$819$	1.26507	0.0442050
$820$	0	0
$821$	11.3404	0.395784	0.197892	−	0.980224i	$-0.436590\pi$
$821$	11.3404	0.395784	0.197892	+	0.980224i	$0.436590\pi$
$822$	0	0
$823$	−21.4209	−0.746684	−0.373342	−	0.927694i	$-0.621788\pi$
$823$	−21.4209	−0.746684	−0.373342	+	0.927694i	$0.621788\pi$
$824$	0	0
$825$	2.44581	0.0851520
$826$	0	0
$827$	−37.8758	−1.31707	−0.658535	−	0.752550i	$-0.728824\pi$
$827$	−37.8758	−1.31707	−0.658535	+	0.752550i	$0.728824\pi$
$828$	0	0
$829$	−49.2141	−1.70928	−0.854638	−	0.519224i	$-0.826221\pi$
$829$	−49.2141	−1.70928	−0.854638	+	0.519224i	$0.826221\pi$
$830$	0	0
$831$	−7.63687	−0.264920
$832$	0	0
$833$	4.90014	0.169780
$834$	0	0
$835$	7.78550	0.269428
$836$	0	0
$837$	21.1292	0.730331
$838$	0	0
$839$	12.5328	0.432681	0.216340	−	0.976318i	$-0.430588\pi$
$839$	12.5328	0.432681	0.216340	+	0.976318i	$0.430588\pi$
$840$	0	0
$841$	15.9003	0.548286
$842$	0	0
$843$	38.5969	1.32935
$844$	0	0
$845$	−13.8879	−0.477758
$846$	0	0
$847$	15.7883	0.542494
$848$	0	0
$849$	−9.32987	−0.320200
$850$	0	0
$851$	−17.5437	−0.601392
$852$	0	0
$853$	8.11152	0.277733	0.138867	−	0.990311i	$-0.455654\pi$
$853$	8.11152	0.277733	0.138867	+	0.990311i	$0.455654\pi$
$854$	0	0
$855$	5.31961	0.181927
$856$	0	0
$857$	−7.01619	−0.239668	−0.119834	−	0.992794i	$-0.538236\pi$
$857$	−7.01619	−0.239668	−0.119834	+	0.992794i	$0.538236\pi$
$858$	0	0
$859$	12.9653	0.442369	0.221184	−	0.975232i	$-0.429008\pi$
$859$	12.9653	0.442369	0.221184	+	0.975232i	$0.429008\pi$
$860$	0	0
$861$	19.6657	0.670205
$862$	0	0
$863$	−44.0282	−1.49874	−0.749369	−	0.662153i	$-0.769643\pi$
$863$	−44.0282	−1.49874	−0.749369	+	0.662153i	$0.769643\pi$
$864$	0	0
$865$	−11.7335	−0.398951
$866$	0	0
$867$	−2.02290	−0.0687012
$868$	0	0
$869$	3.13953	0.106501
$870$	0	0
$871$	−8.03822	−0.272365
$872$	0	0
$873$	4.73194	0.160152
$874$	0	0
$875$	14.2257	0.480917
$876$	0	0
$877$	51.2990	1.73224	0.866122	−	0.499833i	$-0.166605\pi$
$877$	51.2990	1.73224	0.866122	+	0.499833i	$0.166605\pi$
$878$	0	0
$879$	−23.4891	−0.792269
$880$	0	0
$881$	24.9993	0.842248	0.421124	−	0.907003i	$-0.361636\pi$
$881$	24.9993	0.842248	0.421124	+	0.907003i	$0.361636\pi$
$882$	0	0
$883$	1.59295	0.0536072	0.0268036	−	0.999641i	$-0.491467\pi$
$883$	1.59295	0.0536072	0.0268036	+	0.999641i	$0.491467\pi$
$884$	0	0
$885$	−2.27278	−0.0763986
$886$	0	0
$887$	25.1175	0.843364	0.421682	−	0.906744i	$-0.361440\pi$
$887$	25.1175	0.843364	0.421682	+	0.906744i	$0.361440\pi$
$888$	0	0
$889$	−8.82527	−0.295990
$890$	0	0
$891$	−3.58531	−0.120112
$892$	0	0
$893$	47.9472	1.60449
$894$	0	0
$895$	18.6636	0.623855
$896$	0	0
$897$	−3.38953	−0.113173
$898$	0	0
$899$	−36.6843	−1.22349
$900$	0	0
$901$	9.19339	0.306276
$902$	0	0
$903$	−31.5888	−1.05121
$904$	0	0
$905$	2.15333	0.0715790
$906$	0	0
$907$	29.9296	0.993795	0.496898	−	0.867809i	$-0.334472\pi$
$907$	29.9296	0.993795	0.496898	+	0.867809i	$0.334472\pi$
$908$	0	0
$909$	1.56565	0.0519293
$910$	0	0
$911$	−53.1230	−1.76004	−0.880022	−	0.474932i	$-0.842473\pi$
$911$	−53.1230	−1.76004	−0.880022	+	0.474932i	$0.842473\pi$
$912$	0	0
$913$	4.30991	0.142637
$914$	0	0
$915$	10.9959	0.363513
$916$	0	0
$917$	24.5877	0.811959
$918$	0	0
$919$	−34.4628	−1.13682	−0.568412	−	0.822744i	$-0.692442\pi$
$919$	−34.4628	−1.13682	−0.568412	+	0.822744i	$0.692442\pi$
$920$	0	0
$921$	18.2137	0.600161
$922$	0	0
$923$	8.43268	0.277565
$924$	0	0
$925$	−31.2832	−1.02858
$926$	0	0
$927$	−5.58033	−0.183282
$928$	0	0
$929$	25.4877	0.836225	0.418113	−	0.908395i	$-0.362692\pi$
$929$	25.4877	0.836225	0.418113	+	0.908395i	$0.362692\pi$
$930$	0	0
$931$	−21.2441	−0.696246
$932$	0	0
$933$	−35.4644	−1.16105
$934$	0	0
$935$	−0.363436	−0.0118856
$936$	0	0
$937$	5.03335	0.164432	0.0822162	−	0.996615i	$-0.473800\pi$
$937$	5.03335	0.164432	0.0822162	+	0.996615i	$0.473800\pi$
$938$	0	0
$939$	−49.3477	−1.61040
$940$	0	0
$941$	25.5555	0.833087	0.416543	−	0.909116i	$-0.363241\pi$
$941$	25.5555	0.833087	0.416543	+	0.909116i	$0.363241\pi$
$942$	0	0
$943$	−14.0623	−0.457930
$944$	0	0
$945$	−6.28354	−0.204404
$946$	0	0
$947$	57.3318	1.86303	0.931516	−	0.363700i	$-0.118487\pi$
$947$	57.3318	1.86303	0.931516	+	0.363700i	$0.118487\pi$
$948$	0	0
$949$	4.81910	0.156434
$950$	0	0
$951$	3.80284	0.123315
$952$	0	0
$953$	8.95181	0.289978	0.144989	−	0.989433i	$-0.453685\pi$
$953$	8.95181	0.289978	0.144989	+	0.989433i	$0.453685\pi$
$954$	0	0
$955$	10.0211	0.324275
$956$	0	0
$957$	4.38473	0.141738
$958$	0	0
$959$	−10.1451	−0.327601
$960$	0	0
$961$	−1.02826	−0.0331698
$962$	0	0
$963$	−0.273240	−0.00880504
$964$	0	0
$965$	−22.4470	−0.722595
$966$	0	0
$967$	−33.3398	−1.07213	−0.536067	−	0.844175i	$-0.680091\pi$
$967$	−33.3398	−1.07213	−0.536067	+	0.844175i	$0.680091\pi$
$968$	0	0
$969$	8.77006	0.281735
$970$	0	0
$971$	−40.7863	−1.30890	−0.654448	−	0.756107i	$-0.727099\pi$
$971$	−40.7863	−1.30890	−0.654448	+	0.756107i	$0.727099\pi$
$972$	0	0
$973$	23.5076	0.753621
$974$	0	0
$975$	−6.04405	−0.193565
$976$	0	0
$977$	26.7009	0.854236	0.427118	−	0.904196i	$-0.359529\pi$
$977$	26.7009	0.854236	0.427118	+	0.904196i	$0.359529\pi$
$978$	0	0
$979$	5.19996	0.166192
$980$	0	0
$981$	0.538296	0.0171865
$982$	0	0
$983$	−18.2362	−0.581646	−0.290823	−	0.956777i	$-0.593929\pi$
$983$	−18.2362	−0.581646	−0.290823	+	0.956777i	$0.593929\pi$
$984$	0	0
$985$	−7.56847	−0.241151
$986$	0	0
$987$	32.4192	1.03192
$988$	0	0
$989$	22.5881	0.718259
$990$	0	0
$991$	38.8568	1.23433	0.617163	−	0.786835i	$-0.288282\pi$
$991$	38.8568	1.23433	0.617163	+	0.786835i	$0.288282\pi$
$992$	0	0
$993$	35.4174	1.12394
$994$	0	0
$995$	8.92748	0.283020
$996$	0	0
$997$	4.97462	0.157548	0.0787739	−	0.996893i	$-0.474899\pi$
$997$	4.97462	0.157548	0.0787739	+	0.996893i	$0.474899\pi$
$998$	0	0
$999$	32.3024	1.02200

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.x.1.6	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.x.1.6	✓	22	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.02290 q^{3} +1.12353 q^{5} -1.44909 q^{7} +1.09211 q^{9} +O(q^{10})\)	\(q-2.02290 q^{3} +1.12353 q^{5} -1.44909 q^{7} +1.09211 q^{9} +0.323478 q^{11} -0.799377 q^{13} -2.27278 q^{15} -1.00000 q^{17} +4.33540 q^{19} +2.93136 q^{21} -2.09611 q^{23} -3.73769 q^{25} +3.85946 q^{27} -6.70077 q^{29} +5.47464 q^{31} -0.654363 q^{33} -1.62809 q^{35} +8.36966 q^{37} +1.61706 q^{39} +6.70873 q^{41} -10.7762 q^{43} +1.22702 q^{45} +11.0595 q^{47} -4.90014 q^{49} +2.02290 q^{51} -9.19339 q^{53} +0.363436 q^{55} -8.77006 q^{57} +1.00000 q^{59} -4.83808 q^{61} -1.58257 q^{63} -0.898121 q^{65} +10.0556 q^{67} +4.24022 q^{69} -10.5491 q^{71} -6.02857 q^{73} +7.56096 q^{75} -0.468748 q^{77} +9.70554 q^{79} -11.0836 q^{81} +13.3236 q^{83} -1.12353 q^{85} +13.5550 q^{87} +16.0752 q^{89} +1.15837 q^{91} -11.0746 q^{93} +4.87093 q^{95} +4.33283 q^{97} +0.353275 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more