LMFDB - Embedded newform 4024.2.a.g.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.221515 q^{3} +4.40155 q^{5} +5.16895 q^{7} -2.95093 q^{9} +O(q^{10})$	$q+0.221515 q^{3} +4.40155 q^{5} +5.16895 q^{7} -2.95093 q^{9} +3.46962 q^{11} +0.284636 q^{13} +0.975009 q^{15} +1.31008 q^{17} -7.10888 q^{19} +1.14500 q^{21} -4.02329 q^{23} +14.3737 q^{25} -1.31822 q^{27} -1.44801 q^{29} +4.42114 q^{31} +0.768571 q^{33} +22.7514 q^{35} -11.4487 q^{37} +0.0630512 q^{39} +5.15230 q^{41} +3.72817 q^{43} -12.9887 q^{45} +7.61093 q^{47} +19.7181 q^{49} +0.290201 q^{51} +1.44809 q^{53} +15.2717 q^{55} -1.57472 q^{57} -15.0822 q^{59} +11.3809 q^{61} -15.2532 q^{63} +1.25284 q^{65} -0.972122 q^{67} -0.891218 q^{69} +9.54106 q^{71} -13.1975 q^{73} +3.18398 q^{75} +17.9343 q^{77} +4.95916 q^{79} +8.56079 q^{81} +6.33684 q^{83} +5.76637 q^{85} -0.320757 q^{87} -9.69747 q^{89} +1.47127 q^{91} +0.979347 q^{93} -31.2901 q^{95} -4.58016 q^{97} -10.2386 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * 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
$3$	0.221515	0.127892	0.0639458	−	0.997953i	$-0.479632\pi$
$3$	0.221515	0.127892	0.0639458	+	0.997953i	$0.479632\pi$
$4$	0	0
$5$	4.40155	1.96843	0.984217	−	0.176967i	$-0.0566286\pi$
$5$	4.40155	1.96843	0.984217	+	0.176967i	$0.0566286\pi$
$6$	0	0
$7$	5.16895	1.95368	0.976841	−	0.213969i	$-0.0686390\pi$
$7$	5.16895	1.95368	0.976841	+	0.213969i	$0.0686390\pi$
$8$	0	0
$9$	−2.95093	−0.983644
$10$	0	0
$11$	3.46962	1.04613	0.523064	−	0.852293i	$-0.324789\pi$
$11$	3.46962	1.04613	0.523064	+	0.852293i	$0.324789\pi$
$12$	0	0
$13$	0.284636	0.0789439	0.0394720	−	0.999221i	$-0.487432\pi$
$13$	0.284636	0.0789439	0.0394720	+	0.999221i	$0.487432\pi$
$14$	0	0
$15$	0.975009	0.251746
$16$	0	0
$17$	1.31008	0.317740	0.158870	−	0.987299i	$-0.449215\pi$
$17$	1.31008	0.317740	0.158870	+	0.987299i	$0.449215\pi$
$18$	0	0
$19$	−7.10888	−1.63089	−0.815444	−	0.578835i	$-0.803507\pi$
$19$	−7.10888	−1.63089	−0.815444	+	0.578835i	$0.803507\pi$
$20$	0	0
$21$	1.14500	0.249859
$22$	0	0
$23$	−4.02329	−0.838914	−0.419457	−	0.907775i	$-0.637779\pi$
$23$	−4.02329	−0.838914	−0.419457	+	0.907775i	$0.637779\pi$
$24$	0	0
$25$	14.3737	2.87473
$26$	0	0
$27$	−1.31822	−0.253691
$28$	0	0
$29$	−1.44801	−0.268889	−0.134445	−	0.990921i	$-0.542925\pi$
$29$	−1.44801	−0.268889	−0.134445	+	0.990921i	$0.542925\pi$
$30$	0	0
$31$	4.42114	0.794060	0.397030	−	0.917806i	$-0.370041\pi$
$31$	4.42114	0.794060	0.397030	+	0.917806i	$0.370041\pi$
$32$	0	0
$33$	0.768571	0.133791
$34$	0	0
$35$	22.7514	3.84569
$36$	0	0
$37$	−11.4487	−1.88216	−0.941079	−	0.338188i	$-0.890186\pi$
$37$	−11.4487	−1.88216	−0.941079	+	0.338188i	$0.890186\pi$
$38$	0	0
$39$	0.0630512	0.0100963
$40$	0	0
$41$	5.15230	0.804654	0.402327	−	0.915496i	$-0.368201\pi$
$41$	5.15230	0.804654	0.402327	+	0.915496i	$0.368201\pi$
$42$	0	0
$43$	3.72817	0.568541	0.284271	−	0.958744i	$-0.408249\pi$
$43$	3.72817	0.568541	0.284271	+	0.958744i	$0.408249\pi$
$44$	0	0
$45$	−12.9887	−1.93624
$46$	0	0
$47$	7.61093	1.11017	0.555084	−	0.831794i	$-0.312686\pi$
$47$	7.61093	1.11017	0.555084	+	0.831794i	$0.312686\pi$
$48$	0	0
$49$	19.7181	2.81687
$50$	0	0
$51$	0.290201	0.0406363
$52$	0	0
$53$	1.44809	0.198911	0.0994555	−	0.995042i	$-0.468290\pi$
$53$	1.44809	0.198911	0.0994555	+	0.995042i	$0.468290\pi$
$54$	0	0
$55$	15.2717	2.05923
$56$	0	0
$57$	−1.57472	−0.208577
$58$	0	0
$59$	−15.0822	−1.96354	−0.981771	−	0.190069i	$-0.939129\pi$
$59$	−15.0822	−1.96354	−0.981771	+	0.190069i	$0.939129\pi$
$60$	0	0
$61$	11.3809	1.45717	0.728584	−	0.684956i	$-0.240179\pi$
$61$	11.3809	1.45717	0.728584	+	0.684956i	$0.240179\pi$
$62$	0	0
$63$	−15.2532	−1.92173
$64$	0	0
$65$	1.25284	0.155396
$66$	0	0
$67$	−0.972122	−0.118764	−0.0593818	−	0.998235i	$-0.518913\pi$
$67$	−0.972122	−0.118764	−0.0593818	+	0.998235i	$0.518913\pi$
$68$	0	0
$69$	−0.891218	−0.107290
$70$	0	0
$71$	9.54106	1.13232	0.566158	−	0.824297i	$-0.308429\pi$
$71$	9.54106	1.13232	0.566158	+	0.824297i	$0.308429\pi$
$72$	0	0
$73$	−13.1975	−1.54465	−0.772325	−	0.635228i	$-0.780906\pi$
$73$	−13.1975	−1.54465	−0.772325	+	0.635228i	$0.780906\pi$
$74$	0	0
$75$	3.18398	0.367654
$76$	0	0
$77$	17.9343	2.04380
$78$	0	0
$79$	4.95916	0.557949	0.278974	−	0.960299i	$-0.410006\pi$
$79$	4.95916	0.557949	0.278974	+	0.960299i	$0.410006\pi$
$80$	0	0
$81$	8.56079	0.951199
$82$	0	0
$83$	6.33684	0.695559	0.347779	−	0.937576i	$-0.386936\pi$
$83$	6.33684	0.695559	0.347779	+	0.937576i	$0.386936\pi$
$84$	0	0
$85$	5.76637	0.625451
$86$	0	0
$87$	−0.320757	−0.0343887
$88$	0	0
$89$	−9.69747	−1.02793	−0.513965	−	0.857811i	$-0.671824\pi$
$89$	−9.69747	−1.02793	−0.513965	+	0.857811i	$0.671824\pi$
$90$	0	0
$91$	1.47127	0.154231
$92$	0	0
$93$	0.979347	0.101554
$94$	0	0
$95$	−31.2901	−3.21030
$96$	0	0
$97$	−4.58016	−0.465045	−0.232522	−	0.972591i	$-0.574698\pi$
$97$	−4.58016	−0.465045	−0.232522	+	0.972591i	$0.574698\pi$
$98$	0	0
$99$	−10.2386	−1.02902
$100$	0	0
$101$	−4.34743	−0.432585	−0.216293	−	0.976329i	$-0.569397\pi$
$101$	−4.34743	−0.432585	−0.216293	+	0.976329i	$0.569397\pi$
$102$	0	0
$103$	11.3569	1.11903	0.559513	−	0.828822i	$-0.310988\pi$
$103$	11.3569	1.11903	0.559513	+	0.828822i	$0.310988\pi$
$104$	0	0
$105$	5.03978	0.491832
$106$	0	0
$107$	−0.266245	−0.0257389	−0.0128694	−	0.999917i	$-0.504097\pi$
$107$	−0.266245	−0.0257389	−0.0128694	+	0.999917i	$0.504097\pi$
$108$	0	0
$109$	−12.0199	−1.15130	−0.575649	−	0.817697i	$-0.695251\pi$
$109$	−12.0199	−1.15130	−0.575649	+	0.817697i	$0.695251\pi$
$110$	0	0
$111$	−2.53606	−0.240712
$112$	0	0
$113$	−16.1848	−1.52254	−0.761268	−	0.648438i	$-0.775423\pi$
$113$	−16.1848	−1.52254	−0.761268	+	0.648438i	$0.775423\pi$
$114$	0	0
$115$	−17.7087	−1.65135
$116$	0	0
$117$	−0.839942	−0.0776527
$118$	0	0
$119$	6.77173	0.620763
$120$	0	0
$121$	1.03824	0.0943850
$122$	0	0
$123$	1.14131	0.102909
$124$	0	0
$125$	41.2586	3.69028
$126$	0	0
$127$	13.8875	1.23231	0.616156	−	0.787624i	$-0.288689\pi$
$127$	13.8875	1.23231	0.616156	+	0.787624i	$0.288689\pi$
$128$	0	0
$129$	0.825846	0.0727117
$130$	0	0
$131$	−7.58164	−0.662411	−0.331206	−	0.943559i	$-0.607455\pi$
$131$	−7.58164	−0.662411	−0.331206	+	0.943559i	$0.607455\pi$
$132$	0	0
$133$	−36.7455	−3.18624
$134$	0	0
$135$	−5.80221	−0.499375
$136$	0	0
$137$	−10.0247	−0.856468	−0.428234	−	0.903668i	$-0.640864\pi$
$137$	−10.0247	−0.856468	−0.428234	+	0.903668i	$0.640864\pi$
$138$	0	0
$139$	−0.537758	−0.0456120	−0.0228060	−	0.999740i	$-0.507260\pi$
$139$	−0.537758	−0.0456120	−0.0228060	+	0.999740i	$0.507260\pi$
$140$	0	0
$141$	1.68593	0.141981
$142$	0	0
$143$	0.987579	0.0825855
$144$	0	0
$145$	−6.37351	−0.529291
$146$	0	0
$147$	4.36785	0.360254
$148$	0	0
$149$	6.76917	0.554552	0.277276	−	0.960790i	$-0.410568\pi$
$149$	6.76917	0.554552	0.277276	+	0.960790i	$0.410568\pi$
$150$	0	0
$151$	−3.73282	−0.303773	−0.151886	−	0.988398i	$-0.548535\pi$
$151$	−3.73282	−0.303773	−0.151886	+	0.988398i	$0.548535\pi$
$152$	0	0
$153$	−3.86595	−0.312543
$154$	0	0
$155$	19.4599	1.56305
$156$	0	0
$157$	8.55966	0.683135	0.341568	−	0.939857i	$-0.389042\pi$
$157$	8.55966	0.683135	0.341568	+	0.939857i	$0.389042\pi$
$158$	0	0
$159$	0.320774	0.0254390
$160$	0	0
$161$	−20.7962	−1.63897
$162$	0	0
$163$	−2.80202	−0.219471	−0.109736	−	0.993961i	$-0.535000\pi$
$163$	−2.80202	−0.219471	−0.109736	+	0.993961i	$0.535000\pi$
$164$	0	0
$165$	3.38291	0.263359
$166$	0	0
$167$	−8.48792	−0.656815	−0.328407	−	0.944536i	$-0.606512\pi$
$167$	−8.48792	−0.656815	−0.328407	+	0.944536i	$0.606512\pi$
$168$	0	0
$169$	−12.9190	−0.993768
$170$	0	0
$171$	20.9778	1.60421
$172$	0	0
$173$	−3.74290	−0.284567	−0.142284	−	0.989826i	$-0.545445\pi$
$173$	−3.74290	−0.284567	−0.142284	+	0.989826i	$0.545445\pi$
$174$	0	0
$175$	74.2968	5.61631
$176$	0	0
$177$	−3.34094	−0.251121
$178$	0	0
$179$	10.0165	0.748666	0.374333	−	0.927294i	$-0.377872\pi$
$179$	10.0165	0.748666	0.374333	+	0.927294i	$0.377872\pi$
$180$	0	0
$181$	−12.4920	−0.928527	−0.464263	−	0.885697i	$-0.653681\pi$
$181$	−12.4920	−0.928527	−0.464263	+	0.885697i	$0.653681\pi$
$182$	0	0
$183$	2.52103	0.186360
$184$	0	0
$185$	−50.3921	−3.70490
$186$	0	0
$187$	4.54547	0.332397
$188$	0	0
$189$	−6.81382	−0.495632
$190$	0	0
$191$	−10.2645	−0.742711	−0.371355	−	0.928491i	$-0.621107\pi$
$191$	−10.2645	−0.742711	−0.371355	+	0.928491i	$0.621107\pi$
$192$	0	0
$193$	−20.6862	−1.48903	−0.744513	−	0.667608i	$-0.767318\pi$
$193$	−20.6862	−1.48903	−0.744513	+	0.667608i	$0.767318\pi$
$194$	0	0
$195$	0.277523	0.0198738
$196$	0	0
$197$	−6.89634	−0.491344	−0.245672	−	0.969353i	$-0.579009\pi$
$197$	−6.89634	−0.491344	−0.245672	+	0.969353i	$0.579009\pi$
$198$	0	0
$199$	2.58259	0.183075	0.0915376	−	0.995802i	$-0.470822\pi$
$199$	2.58259	0.183075	0.0915376	+	0.995802i	$0.470822\pi$
$200$	0	0
$201$	−0.215339	−0.0151889
$202$	0	0
$203$	−7.48472	−0.525324
$204$	0	0
$205$	22.6781	1.58391
$206$	0	0
$207$	11.8724	0.825192
$208$	0	0
$209$	−24.6651	−1.70612
$210$	0	0
$211$	−2.64402	−0.182022	−0.0910108	−	0.995850i	$-0.529010\pi$
$211$	−2.64402	−0.182022	−0.0910108	+	0.995850i	$0.529010\pi$
$212$	0	0
$213$	2.11349	0.144814
$214$	0	0
$215$	16.4097	1.11914
$216$	0	0
$217$	22.8527	1.55134
$218$	0	0
$219$	−2.92344	−0.197548
$220$	0	0
$221$	0.372896	0.0250837
$222$	0	0
$223$	20.0559	1.34304	0.671522	−	0.740985i	$-0.265641\pi$
$223$	20.0559	1.34304	0.671522	+	0.740985i	$0.265641\pi$
$224$	0	0
$225$	−42.4157	−2.82771
$226$	0	0
$227$	−4.97787	−0.330393	−0.165196	−	0.986261i	$-0.552826\pi$
$227$	−4.97787	−0.330393	−0.165196	+	0.986261i	$0.552826\pi$
$228$	0	0
$229$	11.5503	0.763268	0.381634	−	0.924313i	$-0.375361\pi$
$229$	11.5503	0.763268	0.381634	+	0.924313i	$0.375361\pi$
$230$	0	0
$231$	3.97271	0.261385
$232$	0	0
$233$	−8.14140	−0.533361	−0.266681	−	0.963785i	$-0.585927\pi$
$233$	−8.14140	−0.533361	−0.266681	+	0.963785i	$0.585927\pi$
$234$	0	0
$235$	33.4999	2.18529
$236$	0	0
$237$	1.09853	0.0713570
$238$	0	0
$239$	22.4067	1.44937	0.724684	−	0.689081i	$-0.241986\pi$
$239$	22.4067	1.44937	0.724684	+	0.689081i	$0.241986\pi$
$240$	0	0
$241$	−11.2514	−0.724767	−0.362383	−	0.932029i	$-0.618037\pi$
$241$	−11.2514	−0.724767	−0.362383	+	0.932029i	$0.618037\pi$
$242$	0	0
$243$	5.85100	0.375342
$244$	0	0
$245$	86.7902	5.54482
$246$	0	0
$247$	−2.02345	−0.128749
$248$	0	0
$249$	1.40370	0.0889561
$250$	0	0
$251$	23.9835	1.51383	0.756913	−	0.653516i	$-0.226707\pi$
$251$	23.9835	1.51383	0.756913	+	0.653516i	$0.226707\pi$
$252$	0	0
$253$	−13.9593	−0.877611
$254$	0	0
$255$	1.27734	0.0799899
$256$	0	0
$257$	12.1048	0.755076	0.377538	−	0.925994i	$-0.376771\pi$
$257$	12.1048	0.755076	0.377538	+	0.925994i	$0.376771\pi$
$258$	0	0
$259$	−59.1779	−3.67713
$260$	0	0
$261$	4.27299	0.264491
$262$	0	0
$263$	20.9837	1.29391	0.646954	−	0.762529i	$-0.276043\pi$
$263$	20.9837	1.29391	0.646954	+	0.762529i	$0.276043\pi$
$264$	0	0
$265$	6.37386	0.391543
$266$	0	0
$267$	−2.14813	−0.131464
$268$	0	0
$269$	−24.0181	−1.46441	−0.732204	−	0.681086i	$-0.761508\pi$
$269$	−24.0181	−1.46441	−0.732204	+	0.681086i	$0.761508\pi$
$270$	0	0
$271$	−14.8631	−0.902871	−0.451436	−	0.892304i	$-0.649088\pi$
$271$	−14.8631	−0.902871	−0.451436	+	0.892304i	$0.649088\pi$
$272$	0	0
$273$	0.325909	0.0197249
$274$	0	0
$275$	49.8711	3.00734
$276$	0	0
$277$	−9.20536	−0.553096	−0.276548	−	0.961000i	$-0.589191\pi$
$277$	−9.20536	−0.553096	−0.276548	+	0.961000i	$0.589191\pi$
$278$	0	0
$279$	−13.0465	−0.781072
$280$	0	0
$281$	−9.59880	−0.572616	−0.286308	−	0.958138i	$-0.592428\pi$
$281$	−9.59880	−0.572616	−0.286308	+	0.958138i	$0.592428\pi$
$282$	0	0
$283$	−16.6801	−0.991529	−0.495765	−	0.868457i	$-0.665112\pi$
$283$	−16.6801	−0.991529	−0.495765	+	0.868457i	$0.665112\pi$
$284$	0	0
$285$	−6.93122	−0.410570
$286$	0	0
$287$	26.6320	1.57204
$288$	0	0
$289$	−15.2837	−0.899041
$290$	0	0
$291$	−1.01457	−0.0594754
$292$	0	0
$293$	−15.0456	−0.878975	−0.439488	−	0.898249i	$-0.644840\pi$
$293$	−15.0456	−0.878975	−0.439488	+	0.898249i	$0.644840\pi$
$294$	0	0
$295$	−66.3853	−3.86510
$296$	0	0
$297$	−4.57371	−0.265394
$298$	0	0
$299$	−1.14517	−0.0662271
$300$	0	0
$301$	19.2708	1.11075
$302$	0	0
$303$	−0.963020	−0.0553240
$304$	0	0
$305$	50.0934	2.86834
$306$	0	0
$307$	−22.1072	−1.26172	−0.630862	−	0.775895i	$-0.717299\pi$
$307$	−22.1072	−1.26172	−0.630862	+	0.775895i	$0.717299\pi$
$308$	0	0
$309$	2.51572	0.143114
$310$	0	0
$311$	−10.5560	−0.598577	−0.299288	−	0.954163i	$-0.596749\pi$
$311$	−10.5560	−0.598577	−0.299288	+	0.954163i	$0.596749\pi$
$312$	0	0
$313$	16.7622	0.947457	0.473728	−	0.880671i	$-0.342908\pi$
$313$	16.7622	0.947457	0.473728	+	0.880671i	$0.342908\pi$
$314$	0	0
$315$	−67.1379	−3.78279
$316$	0	0
$317$	−5.95484	−0.334457	−0.167229	−	0.985918i	$-0.553482\pi$
$317$	−5.95484	−0.334457	−0.167229	+	0.985918i	$0.553482\pi$
$318$	0	0
$319$	−5.02405	−0.281293
$320$	0	0
$321$	−0.0589772	−0.00329178
$322$	0	0
$323$	−9.31318	−0.518199
$324$	0	0
$325$	4.09126	0.226943
$326$	0	0
$327$	−2.66259	−0.147241
$328$	0	0
$329$	39.3406	2.16892
$330$	0	0
$331$	−15.3601	−0.844265	−0.422132	−	0.906534i	$-0.638718\pi$
$331$	−15.3601	−0.844265	−0.422132	+	0.906534i	$0.638718\pi$
$332$	0	0
$333$	33.7844	1.85137
$334$	0	0
$335$	−4.27884	−0.233778
$336$	0	0
$337$	12.8826	0.701762	0.350881	−	0.936420i	$-0.385882\pi$
$337$	12.8826	0.701762	0.350881	+	0.936420i	$0.385882\pi$
$338$	0	0
$339$	−3.58517	−0.194719
$340$	0	0
$341$	15.3397	0.830689
$342$	0	0
$343$	65.7392	3.54958
$344$	0	0
$345$	−3.92274	−0.211193
$346$	0	0
$347$	−1.33073	−0.0714375	−0.0357188	−	0.999362i	$-0.511372\pi$
$347$	−1.33073	−0.0714375	−0.0357188	+	0.999362i	$0.511372\pi$
$348$	0	0
$349$	20.3499	1.08930	0.544652	−	0.838662i	$-0.316662\pi$
$349$	20.3499	1.08930	0.544652	+	0.838662i	$0.316662\pi$
$350$	0	0
$351$	−0.375213	−0.0200274
$352$	0	0
$353$	−2.64533	−0.140797	−0.0703984	−	0.997519i	$-0.522427\pi$
$353$	−2.64533	−0.140797	−0.0703984	+	0.997519i	$0.522427\pi$
$354$	0	0
$355$	41.9955	2.22889
$356$	0	0
$357$	1.50004	0.0793904
$358$	0	0
$359$	−28.3784	−1.49775	−0.748877	−	0.662709i	$-0.769407\pi$
$359$	−28.3784	−1.49775	−0.748877	+	0.662709i	$0.769407\pi$
$360$	0	0
$361$	31.5362	1.65980
$362$	0	0
$363$	0.229985	0.0120711
$364$	0	0
$365$	−58.0894	−3.04054
$366$	0	0
$367$	−13.9810	−0.729804	−0.364902	−	0.931046i	$-0.618897\pi$
$367$	−13.9810	−0.729804	−0.364902	+	0.931046i	$0.618897\pi$
$368$	0	0
$369$	−15.2041	−0.791493
$370$	0	0
$371$	7.48513	0.388609
$372$	0	0
$373$	14.1713	0.733764	0.366882	−	0.930268i	$-0.380425\pi$
$373$	14.1713	0.733764	0.366882	+	0.930268i	$0.380425\pi$
$374$	0	0
$375$	9.13939	0.471956
$376$	0	0
$377$	−0.412157	−0.0212272
$378$	0	0
$379$	29.3714	1.50871	0.754354	−	0.656468i	$-0.227950\pi$
$379$	29.3714	1.50871	0.754354	+	0.656468i	$0.227950\pi$
$380$	0	0
$381$	3.07628	0.157602
$382$	0	0
$383$	19.9090	1.01730	0.508650	−	0.860973i	$-0.330145\pi$
$383$	19.9090	1.01730	0.508650	+	0.860973i	$0.330145\pi$
$384$	0	0
$385$	78.9387	4.02309
$386$	0	0
$387$	−11.0016	−0.559242
$388$	0	0
$389$	4.59306	0.232877	0.116439	−	0.993198i	$-0.462852\pi$
$389$	4.59306	0.232877	0.116439	+	0.993198i	$0.462852\pi$
$390$	0	0
$391$	−5.27082	−0.266557
$392$	0	0
$393$	−1.67945	−0.0847168
$394$	0	0
$395$	21.8280	1.09828
$396$	0	0
$397$	28.9361	1.45226	0.726131	−	0.687557i	$-0.241317\pi$
$397$	28.9361	1.45226	0.726131	+	0.687557i	$0.241317\pi$
$398$	0	0
$399$	−8.13967	−0.407493
$400$	0	0
$401$	32.6811	1.63202	0.816008	−	0.578041i	$-0.196183\pi$
$401$	32.6811	1.63202	0.816008	+	0.578041i	$0.196183\pi$
$402$	0	0
$403$	1.25842	0.0626862
$404$	0	0
$405$	37.6807	1.87237
$406$	0	0
$407$	−39.7226	−1.96898
$408$	0	0
$409$	−16.9053	−0.835914	−0.417957	−	0.908467i	$-0.637254\pi$
$409$	−16.9053	−0.835914	−0.417957	+	0.908467i	$0.637254\pi$
$410$	0	0
$411$	−2.22062	−0.109535
$412$	0	0
$413$	−77.9595	−3.83613
$414$	0	0
$415$	27.8919	1.36916
$416$	0	0
$417$	−0.119121	−0.00583339
$418$	0	0
$419$	−18.4215	−0.899949	−0.449975	−	0.893041i	$-0.648567\pi$
$419$	−18.4215	−0.899949	−0.449975	+	0.893041i	$0.648567\pi$
$420$	0	0
$421$	−7.30870	−0.356204	−0.178102	−	0.984012i	$-0.556996\pi$
$421$	−7.30870	−0.356204	−0.178102	+	0.984012i	$0.556996\pi$
$422$	0	0
$423$	−22.4593	−1.09201
$424$	0	0
$425$	18.8306	0.913418
$426$	0	0
$427$	58.8271	2.84684
$428$	0	0
$429$	0.218763	0.0105620
$430$	0	0
$431$	28.0506	1.35115	0.675575	−	0.737292i	$-0.263896\pi$
$431$	28.0506	1.35115	0.675575	+	0.737292i	$0.263896\pi$
$432$	0	0
$433$	−8.23781	−0.395884	−0.197942	−	0.980214i	$-0.563426\pi$
$433$	−8.23781	−0.395884	−0.197942	+	0.980214i	$0.563426\pi$
$434$	0	0
$435$	−1.41183	−0.0676919
$436$	0	0
$437$	28.6011	1.36817
$438$	0	0
$439$	7.72651	0.368766	0.184383	−	0.982854i	$-0.440971\pi$
$439$	7.72651	0.368766	0.184383	+	0.982854i	$0.440971\pi$
$440$	0	0
$441$	−58.1867	−2.77080
$442$	0	0
$443$	16.4460	0.781373	0.390686	−	0.920524i	$-0.372238\pi$
$443$	16.4460	0.781373	0.390686	+	0.920524i	$0.372238\pi$
$444$	0	0
$445$	−42.6839	−2.02341
$446$	0	0
$447$	1.49947	0.0709225
$448$	0	0
$449$	−11.2681	−0.531773	−0.265887	−	0.964004i	$-0.585665\pi$
$449$	−11.2681	−0.531773	−0.265887	+	0.964004i	$0.585665\pi$
$450$	0	0
$451$	17.8765	0.841772
$452$	0	0
$453$	−0.826875	−0.0388500
$454$	0	0
$455$	6.47588	0.303594
$456$	0	0
$457$	32.6461	1.52712	0.763561	−	0.645736i	$-0.223449\pi$
$457$	32.6461	1.52712	0.763561	+	0.645736i	$0.223449\pi$
$458$	0	0
$459$	−1.72697	−0.0806080
$460$	0	0
$461$	−16.0607	−0.748021	−0.374011	−	0.927424i	$-0.622018\pi$
$461$	−16.0607	−0.748021	−0.374011	+	0.927424i	$0.622018\pi$
$462$	0	0
$463$	19.4868	0.905627	0.452814	−	0.891605i	$-0.350420\pi$
$463$	19.4868	0.905627	0.452814	+	0.891605i	$0.350420\pi$
$464$	0	0
$465$	4.31065	0.199902
$466$	0	0
$467$	7.24036	0.335044	0.167522	−	0.985868i	$-0.446423\pi$
$467$	7.24036	0.335044	0.167522	+	0.985868i	$0.446423\pi$
$468$	0	0
$469$	−5.02485	−0.232026
$470$	0	0
$471$	1.89609	0.0873673
$472$	0	0
$473$	12.9353	0.594767
$474$	0	0
$475$	−102.181	−4.68837
$476$	0	0
$477$	−4.27322	−0.195657
$478$	0	0
$479$	23.5795	1.07738	0.538688	−	0.842506i	$-0.318920\pi$
$479$	23.5795	1.07738	0.538688	+	0.842506i	$0.318920\pi$
$480$	0	0
$481$	−3.25872	−0.148585
$482$	0	0
$483$	−4.60666	−0.209610
$484$	0	0
$485$	−20.1598	−0.915410
$486$	0	0
$487$	−3.50273	−0.158724	−0.0793619	−	0.996846i	$-0.525288\pi$
$487$	−3.50273	−0.158724	−0.0793619	+	0.996846i	$0.525288\pi$
$488$	0	0
$489$	−0.620690	−0.0280686
$490$	0	0
$491$	26.9730	1.21728	0.608638	−	0.793448i	$-0.291716\pi$
$491$	26.9730	1.21728	0.608638	+	0.793448i	$0.291716\pi$
$492$	0	0
$493$	−1.89701	−0.0854370
$494$	0	0
$495$	−45.0657	−2.02555
$496$	0	0
$497$	49.3173	2.21218
$498$	0	0
$499$	−37.0300	−1.65769	−0.828846	−	0.559477i	$-0.811002\pi$
$499$	−37.0300	−1.65769	−0.828846	+	0.559477i	$0.811002\pi$
$500$	0	0
$501$	−1.88020	−0.0840011
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−19.1354	−0.851515
$506$	0	0
$507$	−2.86175	−0.127095
$508$	0	0
$509$	−38.1818	−1.69238	−0.846189	−	0.532884i	$-0.821108\pi$
$509$	−38.1818	−1.69238	−0.846189	+	0.532884i	$0.821108\pi$
$510$	0	0
$511$	−68.2172	−3.01775
$512$	0	0
$513$	9.37106	0.413743
$514$	0	0
$515$	49.9879	2.20273
$516$	0	0
$517$	26.4070	1.16138
$518$	0	0
$519$	−0.829108	−0.0363938
$520$	0	0
$521$	7.55654	0.331058	0.165529	−	0.986205i	$-0.447067\pi$
$521$	7.55654	0.331058	0.165529	+	0.986205i	$0.447067\pi$
$522$	0	0
$523$	21.4990	0.940084	0.470042	−	0.882644i	$-0.344239\pi$
$523$	21.4990	0.940084	0.470042	+	0.882644i	$0.344239\pi$
$524$	0	0
$525$	16.4578	0.718279
$526$	0	0
$527$	5.79203	0.252305
$528$	0	0
$529$	−6.81315	−0.296224
$530$	0	0
$531$	44.5067	1.93143
$532$	0	0
$533$	1.46653	0.0635225
$534$	0	0
$535$	−1.17189	−0.0506652
$536$	0	0
$537$	2.21879	0.0957481
$538$	0	0
$539$	68.4142	2.94681
$540$	0	0
$541$	37.7447	1.62277	0.811386	−	0.584511i	$-0.198714\pi$
$541$	37.7447	1.62277	0.811386	+	0.584511i	$0.198714\pi$
$542$	0	0
$543$	−2.76717	−0.118751
$544$	0	0
$545$	−52.9062	−2.26625
$546$	0	0
$547$	5.55779	0.237634	0.118817	−	0.992916i	$-0.462090\pi$
$547$	5.55779	0.237634	0.118817	+	0.992916i	$0.462090\pi$
$548$	0	0
$549$	−33.5841	−1.43333
$550$	0	0
$551$	10.2938	0.438529
$552$	0	0
$553$	25.6336	1.09005
$554$	0	0
$555$	−11.1626	−0.473826
$556$	0	0
$557$	−22.9853	−0.973917	−0.486958	−	0.873425i	$-0.661894\pi$
$557$	−22.9853	−0.973917	−0.486958	+	0.873425i	$0.661894\pi$
$558$	0	0
$559$	1.06117	0.0448829
$560$	0	0
$561$	1.00689	0.0425108
$562$	0	0
$563$	−25.1024	−1.05794	−0.528970	−	0.848640i	$-0.677422\pi$
$563$	−25.1024	−1.05794	−0.528970	+	0.848640i	$0.677422\pi$
$564$	0	0
$565$	−71.2381	−2.99701
$566$	0	0
$567$	44.2503	1.85834
$568$	0	0
$569$	−19.1286	−0.801912	−0.400956	−	0.916097i	$-0.631322\pi$
$569$	−19.1286	−0.801912	−0.400956	+	0.916097i	$0.631322\pi$
$570$	0	0
$571$	−2.89793	−0.121275	−0.0606373	−	0.998160i	$-0.519313\pi$
$571$	−2.89793	−0.121275	−0.0606373	+	0.998160i	$0.519313\pi$
$572$	0	0
$573$	−2.27373	−0.0949865
$574$	0	0
$575$	−57.8294	−2.41165
$576$	0	0
$577$	−7.28821	−0.303412	−0.151706	−	0.988426i	$-0.548477\pi$
$577$	−7.28821	−0.303412	−0.151706	+	0.988426i	$0.548477\pi$
$578$	0	0
$579$	−4.58230	−0.190434
$580$	0	0
$581$	32.7548	1.35890
$582$	0	0
$583$	5.02433	0.208086
$584$	0	0
$585$	−3.69705	−0.152854
$586$	0	0
$587$	−15.6488	−0.645896	−0.322948	−	0.946417i	$-0.604674\pi$
$587$	−15.6488	−0.645896	−0.322948	+	0.946417i	$0.604674\pi$
$588$	0	0
$589$	−31.4293	−1.29502
$590$	0	0
$591$	−1.52764	−0.0628387
$592$	0	0
$593$	21.3533	0.876873	0.438437	−	0.898762i	$-0.355532\pi$
$593$	21.3533	0.876873	0.438437	+	0.898762i	$0.355532\pi$
$594$	0	0
$595$	29.8061	1.22193
$596$	0	0
$597$	0.572082	0.0234138
$598$	0	0
$599$	−42.6394	−1.74220	−0.871099	−	0.491107i	$-0.836592\pi$
$599$	−42.6394	−1.74220	−0.871099	+	0.491107i	$0.836592\pi$
$600$	0	0
$601$	−21.1371	−0.862200	−0.431100	−	0.902304i	$-0.641874\pi$
$601$	−21.1371	−0.862200	−0.431100	+	0.902304i	$0.641874\pi$
$602$	0	0
$603$	2.86866	0.116821
$604$	0	0
$605$	4.56985	0.185791
$606$	0	0
$607$	−26.4270	−1.07264	−0.536320	−	0.844015i	$-0.680186\pi$
$607$	−26.4270	−1.07264	−0.536320	+	0.844015i	$0.680186\pi$
$608$	0	0
$609$	−1.65798	−0.0671846
$610$	0	0
$611$	2.16635	0.0876411
$612$	0	0
$613$	−1.43920	−0.0581285	−0.0290643	−	0.999578i	$-0.509253\pi$
$613$	−1.43920	−0.0581285	−0.0290643	+	0.999578i	$0.509253\pi$
$614$	0	0
$615$	5.02354	0.202569
$616$	0	0
$617$	−23.7887	−0.957697	−0.478848	−	0.877898i	$-0.658946\pi$
$617$	−23.7887	−0.957697	−0.478848	+	0.877898i	$0.658946\pi$
$618$	0	0
$619$	−28.1966	−1.13332	−0.566659	−	0.823953i	$-0.691764\pi$
$619$	−28.1966	−1.13332	−0.566659	+	0.823953i	$0.691764\pi$
$620$	0	0
$621$	5.30358	0.212825
$622$	0	0
$623$	−50.1258	−2.00825
$624$	0	0
$625$	109.734	4.38935
$626$	0	0
$627$	−5.46368	−0.218198
$628$	0	0
$629$	−14.9987	−0.598037
$630$	0	0
$631$	−40.1907	−1.59997	−0.799983	−	0.600023i	$-0.795158\pi$
$631$	−40.1907	−1.59997	−0.799983	+	0.600023i	$0.795158\pi$
$632$	0	0
$633$	−0.585689	−0.0232790
$634$	0	0
$635$	61.1263	2.42572
$636$	0	0
$637$	5.61248	0.222375
$638$	0	0
$639$	−28.1550	−1.11380
$640$	0	0
$641$	−31.9857	−1.26336	−0.631679	−	0.775230i	$-0.717634\pi$
$641$	−31.9857	−1.26336	−0.631679	+	0.775230i	$0.717634\pi$
$642$	0	0
$643$	−40.2516	−1.58737	−0.793684	−	0.608330i	$-0.791840\pi$
$643$	−40.2516	−1.58737	−0.793684	+	0.608330i	$0.791840\pi$
$644$	0	0
$645$	3.63500	0.143128
$646$	0	0
$647$	16.4669	0.647382	0.323691	−	0.946163i	$-0.395076\pi$
$647$	16.4669	0.647382	0.323691	+	0.946163i	$0.395076\pi$
$648$	0	0
$649$	−52.3296	−2.05412
$650$	0	0
$651$	5.06220	0.198403
$652$	0	0
$653$	42.5392	1.66469	0.832343	−	0.554261i	$-0.186999\pi$
$653$	42.5392	1.66469	0.832343	+	0.554261i	$0.186999\pi$
$654$	0	0
$655$	−33.3710	−1.30391
$656$	0	0
$657$	38.9449	1.51938
$658$	0	0
$659$	−40.2058	−1.56619	−0.783097	−	0.621899i	$-0.786361\pi$
$659$	−40.2058	−1.56619	−0.783097	+	0.621899i	$0.786361\pi$
$660$	0	0
$661$	−35.9376	−1.39781	−0.698905	−	0.715214i	$-0.746329\pi$
$661$	−35.9376	−1.39781	−0.698905	+	0.715214i	$0.746329\pi$
$662$	0	0
$663$	0.0826019	0.00320799
$664$	0	0
$665$	−161.737	−6.27190
$666$	0	0
$667$	5.82578	0.225575
$668$	0	0
$669$	4.44268	0.171764
$670$	0	0
$671$	39.4872	1.52439
$672$	0	0
$673$	−42.9113	−1.65411	−0.827055	−	0.562121i	$-0.809985\pi$
$673$	−42.9113	−1.65411	−0.827055	+	0.562121i	$0.809985\pi$
$674$	0	0
$675$	−18.9476	−0.729295
$676$	0	0
$677$	−14.5634	−0.559718	−0.279859	−	0.960041i	$-0.590288\pi$
$677$	−14.5634	−0.559718	−0.279859	+	0.960041i	$0.590288\pi$
$678$	0	0
$679$	−23.6746	−0.908550
$680$	0	0
$681$	−1.10267	−0.0422544
$682$	0	0
$683$	−11.8152	−0.452096	−0.226048	−	0.974116i	$-0.572581\pi$
$683$	−11.8152	−0.452096	−0.226048	+	0.974116i	$0.572581\pi$
$684$	0	0
$685$	−44.1242	−1.68590
$686$	0	0
$687$	2.55857	0.0976156
$688$	0	0
$689$	0.412180	0.0157028
$690$	0	0
$691$	1.19395	0.0454198	0.0227099	−	0.999742i	$-0.492771\pi$
$691$	1.19395	0.0454198	0.0227099	+	0.999742i	$0.492771\pi$
$692$	0	0
$693$	−52.9228	−2.01037
$694$	0	0
$695$	−2.36697	−0.0897842
$696$	0	0
$697$	6.74991	0.255671
$698$	0	0
$699$	−1.80344	−0.0682124
$700$	0	0
$701$	24.1966	0.913891	0.456946	−	0.889495i	$-0.348943\pi$
$701$	24.1966	0.913891	0.456946	+	0.889495i	$0.348943\pi$
$702$	0	0
$703$	81.3875	3.06959
$704$	0	0
$705$	7.42073	0.279481
$706$	0	0
$707$	−22.4717	−0.845134
$708$	0	0
$709$	8.64330	0.324606	0.162303	−	0.986741i	$-0.448108\pi$
$709$	8.64330	0.324606	0.162303	+	0.986741i	$0.448108\pi$
$710$	0	0
$711$	−14.6341	−0.548823
$712$	0	0
$713$	−17.7875	−0.666148
$714$	0	0
$715$	4.34688	0.162564
$716$	0	0
$717$	4.96341	0.185362
$718$	0	0
$719$	36.7483	1.37048	0.685241	−	0.728317i	$-0.259697\pi$
$719$	36.7483	1.37048	0.685241	+	0.728317i	$0.259697\pi$
$720$	0	0
$721$	58.7032	2.18622
$722$	0	0
$723$	−2.49235	−0.0926916
$724$	0	0
$725$	−20.8133	−0.772985
$726$	0	0
$727$	−21.7236	−0.805682	−0.402841	−	0.915270i	$-0.631977\pi$
$727$	−21.7236	−0.805682	−0.402841	+	0.915270i	$0.631977\pi$
$728$	0	0
$729$	−24.3863	−0.903196
$730$	0	0
$731$	4.88420	0.180649
$732$	0	0
$733$	8.85237	0.326970	0.163485	−	0.986546i	$-0.447726\pi$
$733$	8.85237	0.326970	0.163485	+	0.986546i	$0.447726\pi$
$734$	0	0
$735$	19.2253	0.709136
$736$	0	0
$737$	−3.37289	−0.124242
$738$	0	0
$739$	−45.7377	−1.68249	−0.841244	−	0.540656i	$-0.818176\pi$
$739$	−45.7377	−1.68249	−0.841244	+	0.540656i	$0.818176\pi$
$740$	0	0
$741$	−0.448223	−0.0164659
$742$	0	0
$743$	9.17596	0.336633	0.168317	−	0.985733i	$-0.446167\pi$
$743$	9.17596	0.336633	0.168317	+	0.985733i	$0.446167\pi$
$744$	0	0
$745$	29.7948	1.09160
$746$	0	0
$747$	−18.6996	−0.684182
$748$	0	0
$749$	−1.37621	−0.0502855
$750$	0	0
$751$	40.7935	1.48858	0.744288	−	0.667859i	$-0.232789\pi$
$751$	40.7935	1.48858	0.744288	+	0.667859i	$0.232789\pi$
$752$	0	0
$753$	5.31270	0.193606
$754$	0	0
$755$	−16.4302	−0.597956
$756$	0	0
$757$	−43.8350	−1.59321	−0.796606	−	0.604499i	$-0.793373\pi$
$757$	−43.8350	−1.59321	−0.796606	+	0.604499i	$0.793373\pi$
$758$	0	0
$759$	−3.09218	−0.112239
$760$	0	0
$761$	−9.07830	−0.329088	−0.164544	−	0.986370i	$-0.552615\pi$
$761$	−9.07830	−0.329088	−0.164544	+	0.986370i	$0.552615\pi$
$762$	0	0
$763$	−62.1303	−2.24927
$764$	0	0
$765$	−17.0162	−0.615221
$766$	0	0
$767$	−4.29296	−0.155010
$768$	0	0
$769$	17.1580	0.618733	0.309367	−	0.950943i	$-0.399883\pi$
$769$	17.1580	0.618733	0.309367	+	0.950943i	$0.399883\pi$
$770$	0	0
$771$	2.68139	0.0965679
$772$	0	0
$773$	43.9334	1.58018	0.790088	−	0.612994i	$-0.210035\pi$
$773$	43.9334	1.58018	0.790088	+	0.612994i	$0.210035\pi$
$774$	0	0
$775$	63.5479	2.28271
$776$	0	0
$777$	−13.1088	−0.470275
$778$	0	0
$779$	−36.6271	−1.31230
$780$	0	0
$781$	33.1038	1.18455
$782$	0	0
$783$	1.90880	0.0682149
$784$	0	0
$785$	37.6758	1.34471
$786$	0	0
$787$	9.46716	0.337468	0.168734	−	0.985662i	$-0.446032\pi$
$787$	9.46716	0.337468	0.168734	+	0.985662i	$0.446032\pi$
$788$	0	0
$789$	4.64819	0.165480
$790$	0	0
$791$	−83.6583	−2.97455
$792$	0	0
$793$	3.23940	0.115035
$794$	0	0
$795$	1.41190	0.0500751
$796$	0	0
$797$	35.8093	1.26843	0.634215	−	0.773156i	$-0.281323\pi$
$797$	35.8093	1.26843	0.634215	+	0.773156i	$0.281323\pi$
$798$	0	0
$799$	9.97091	0.352745
$800$	0	0
$801$	28.6166	1.01112
$802$	0	0
$803$	−45.7902	−1.61590
$804$	0	0
$805$	−91.5355	−3.22620
$806$	0	0
$807$	−5.32036	−0.187285
$808$	0	0
$809$	11.4204	0.401521	0.200760	−	0.979640i	$-0.435659\pi$
$809$	11.4204	0.401521	0.200760	+	0.979640i	$0.435659\pi$
$810$	0	0
$811$	−13.1818	−0.462877	−0.231438	−	0.972850i	$-0.574343\pi$
$811$	−13.1818	−0.462877	−0.231438	+	0.972850i	$0.574343\pi$
$812$	0	0
$813$	−3.29240	−0.115470
$814$	0	0
$815$	−12.3332	−0.432015
$816$	0	0
$817$	−26.5031	−0.927228
$818$	0	0
$819$	−4.34162	−0.151709
$820$	0	0
$821$	−22.3076	−0.778540	−0.389270	−	0.921124i	$-0.627273\pi$
$821$	−22.3076	−0.778540	−0.389270	+	0.921124i	$0.627273\pi$
$822$	0	0
$823$	−18.8812	−0.658157	−0.329079	−	0.944302i	$-0.606738\pi$
$823$	−18.8812	−0.658157	−0.329079	+	0.944302i	$0.606738\pi$
$824$	0	0
$825$	11.0472	0.384613
$826$	0	0
$827$	20.4117	0.709784	0.354892	−	0.934907i	$-0.384518\pi$
$827$	20.4117	0.709784	0.354892	+	0.934907i	$0.384518\pi$
$828$	0	0
$829$	−21.3411	−0.741207	−0.370604	−	0.928791i	$-0.620849\pi$
$829$	−21.3411	−0.741207	−0.370604	+	0.928791i	$0.620849\pi$
$830$	0	0
$831$	−2.03912	−0.0707364
$832$	0	0
$833$	25.8322	0.895033
$834$	0	0
$835$	−37.3600	−1.29290
$836$	0	0
$837$	−5.82803	−0.201446
$838$	0	0
$839$	40.5057	1.39841	0.699206	−	0.714920i	$-0.253537\pi$
$839$	40.5057	1.39841	0.699206	+	0.714920i	$0.253537\pi$
$840$	0	0
$841$	−26.9033	−0.927698
$842$	0	0
$843$	−2.12628	−0.0732328
$844$	0	0
$845$	−56.8636	−1.95617
$846$	0	0
$847$	5.36659	0.184398
$848$	0	0
$849$	−3.69489	−0.126808
$850$	0	0
$851$	46.0615	1.57897
$852$	0	0
$853$	35.6274	1.21986	0.609930	−	0.792455i	$-0.291197\pi$
$853$	35.6274	1.21986	0.609930	+	0.792455i	$0.291197\pi$
$854$	0	0
$855$	92.3349	3.15779
$856$	0	0
$857$	23.6966	0.809461	0.404730	−	0.914436i	$-0.367365\pi$
$857$	23.6966	0.809461	0.404730	+	0.914436i	$0.367365\pi$
$858$	0	0
$859$	48.6165	1.65877	0.829387	−	0.558675i	$-0.188690\pi$
$859$	48.6165	1.65877	0.829387	+	0.558675i	$0.188690\pi$
$860$	0	0
$861$	5.89938	0.201050
$862$	0	0
$863$	43.8163	1.49152	0.745761	−	0.666213i	$-0.232086\pi$
$863$	43.8163	1.49152	0.745761	+	0.666213i	$0.232086\pi$
$864$	0	0
$865$	−16.4746	−0.560152
$866$	0	0
$867$	−3.38557	−0.114980
$868$	0	0
$869$	17.2064	0.583686
$870$	0	0
$871$	−0.276701	−0.00937566
$872$	0	0
$873$	13.5157	0.457439
$874$	0	0
$875$	213.264	7.20964
$876$	0	0
$877$	7.34486	0.248018	0.124009	−	0.992281i	$-0.460425\pi$
$877$	7.34486	0.248018	0.124009	+	0.992281i	$0.460425\pi$
$878$	0	0
$879$	−3.33283	−0.112414
$880$	0	0
$881$	40.8473	1.37618	0.688090	−	0.725625i	$-0.258449\pi$
$881$	40.8473	1.37618	0.688090	+	0.725625i	$0.258449\pi$
$882$	0	0
$883$	−33.0068	−1.11077	−0.555384	−	0.831594i	$-0.687429\pi$
$883$	−33.0068	−1.11077	−0.555384	+	0.831594i	$0.687429\pi$
$884$	0	0
$885$	−14.7053	−0.494314
$886$	0	0
$887$	−3.07359	−0.103201	−0.0516005	−	0.998668i	$-0.516432\pi$
$887$	−3.07359	−0.103201	−0.0516005	+	0.998668i	$0.516432\pi$
$888$	0	0
$889$	71.7836	2.40755
$890$	0	0
$891$	29.7026	0.995076
$892$	0	0
$893$	−54.1052	−1.81056
$894$	0	0
$895$	44.0880	1.47370
$896$	0	0
$897$	−0.253673	−0.00846989
$898$	0	0
$899$	−6.40187	−0.213514
$900$	0	0
$901$	1.89711	0.0632020
$902$	0	0
$903$	4.26876	0.142055
$904$	0	0
$905$	−54.9844	−1.82774
$906$	0	0
$907$	31.3977	1.04254	0.521272	−	0.853390i	$-0.325458\pi$
$907$	31.3977	1.04254	0.521272	+	0.853390i	$0.325458\pi$
$908$	0	0
$909$	12.8290	0.425510
$910$	0	0
$911$	21.6158	0.716162	0.358081	−	0.933691i	$-0.383431\pi$
$911$	21.6158	0.716162	0.358081	+	0.933691i	$0.383431\pi$
$912$	0	0
$913$	21.9864	0.727644
$914$	0	0
$915$	11.0964	0.366837
$916$	0	0
$917$	−39.1892	−1.29414
$918$	0	0
$919$	35.0110	1.15490	0.577452	−	0.816424i	$-0.304047\pi$
$919$	35.0110	1.15490	0.577452	+	0.816424i	$0.304047\pi$
$920$	0	0
$921$	−4.89707	−0.161364
$922$	0	0
$923$	2.71573	0.0893895
$924$	0	0
$925$	−164.560	−5.41069
$926$	0	0
$927$	−33.5134	−1.10072
$928$	0	0
$929$	1.76321	0.0578490	0.0289245	−	0.999582i	$-0.490792\pi$
$929$	1.76321	0.0578490	0.0289245	+	0.999582i	$0.490792\pi$
$930$	0	0
$931$	−140.174	−4.59400
$932$	0	0
$933$	−2.33831	−0.0765529
$934$	0	0
$935$	20.0071	0.654302
$936$	0	0
$937$	−60.7539	−1.98474	−0.992372	−	0.123283i	$-0.960658\pi$
$937$	−60.7539	−1.98474	−0.992372	+	0.123283i	$0.960658\pi$
$938$	0	0
$939$	3.71308	0.121172
$940$	0	0
$941$	30.3768	0.990256	0.495128	−	0.868820i	$-0.335121\pi$
$941$	30.3768	0.990256	0.495128	+	0.868820i	$0.335121\pi$
$942$	0	0
$943$	−20.7292	−0.675035
$944$	0	0
$945$	−29.9914	−0.975619
$946$	0	0
$947$	−26.9916	−0.877108	−0.438554	−	0.898705i	$-0.644509\pi$
$947$	−26.9916	−0.877108	−0.438554	+	0.898705i	$0.644509\pi$
$948$	0	0
$949$	−3.75649	−0.121941
$950$	0	0
$951$	−1.31909	−0.0427743
$952$	0	0
$953$	−2.07824	−0.0673207	−0.0336603	−	0.999433i	$-0.510716\pi$
$953$	−2.07824	−0.0673207	−0.0336603	+	0.999433i	$0.510716\pi$
$954$	0	0
$955$	−45.1796	−1.46198
$956$	0	0
$957$	−1.11290	−0.0359750
$958$	0	0
$959$	−51.8172	−1.67326
$960$	0	0
$961$	−11.4535	−0.369469
$962$	0	0
$963$	0.785670	0.0253179
$964$	0	0
$965$	−91.0514	−2.93105
$966$	0	0
$967$	5.09031	0.163693	0.0818466	−	0.996645i	$-0.473918\pi$
$967$	5.09031	0.163693	0.0818466	+	0.996645i	$0.473918\pi$
$968$	0	0
$969$	−2.06301	−0.0662734
$970$	0	0
$971$	36.6499	1.17615	0.588075	−	0.808806i	$-0.299886\pi$
$971$	36.6499	1.17615	0.588075	+	0.808806i	$0.299886\pi$
$972$	0	0
$973$	−2.77964	−0.0891113
$974$	0	0
$975$	0.906276	0.0290240
$976$	0	0
$977$	9.76709	0.312477	0.156239	−	0.987719i	$-0.450063\pi$
$977$	9.76709	0.312477	0.156239	+	0.987719i	$0.450063\pi$
$978$	0	0
$979$	−33.6465	−1.07535
$980$	0	0
$981$	35.4699	1.13247
$982$	0	0
$983$	34.4866	1.09995	0.549975	−	0.835181i	$-0.314637\pi$
$983$	34.4866	1.09995	0.549975	+	0.835181i	$0.314637\pi$
$984$	0	0
$985$	−30.3546	−0.967177
$986$	0	0
$987$	8.71452	0.277386
$988$	0	0
$989$	−14.9995	−0.476957
$990$	0	0
$991$	−24.1429	−0.766924	−0.383462	−	0.923557i	$-0.625268\pi$
$991$	−24.1429	−0.766924	−0.383462	+	0.923557i	$0.625268\pi$
$992$	0	0
$993$	−3.40248	−0.107974
$994$	0	0
$995$	11.3674	0.360371
$996$	0	0
$997$	−27.7284	−0.878168	−0.439084	−	0.898446i	$-0.644697\pi$
$997$	−27.7284	−0.878168	−0.439084	+	0.898446i	$0.644697\pi$
$998$	0	0
$999$	15.0919	0.477487

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.g.1.15	✓	33
4.3	odd	2		8048.2.a.x.1.19		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.15	✓	33	1.1	even	1	trivial
8048.2.a.x.1.19		33	4.3	odd	2

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:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q+0.221515 q^{3} +4.40155 q^{5} +5.16895 q^{7} -2.95093 q^{9} +O(q^{10})\)	\(q+0.221515 q^{3} +4.40155 q^{5} +5.16895 q^{7} -2.95093 q^{9} +3.46962 q^{11} +0.284636 q^{13} +0.975009 q^{15} +1.31008 q^{17} -7.10888 q^{19} +1.14500 q^{21} -4.02329 q^{23} +14.3737 q^{25} -1.31822 q^{27} -1.44801 q^{29} +4.42114 q^{31} +0.768571 q^{33} +22.7514 q^{35} -11.4487 q^{37} +0.0630512 q^{39} +5.15230 q^{41} +3.72817 q^{43} -12.9887 q^{45} +7.61093 q^{47} +19.7181 q^{49} +0.290201 q^{51} +1.44809 q^{53} +15.2717 q^{55} -1.57472 q^{57} -15.0822 q^{59} +11.3809 q^{61} -15.2532 q^{63} +1.25284 q^{65} -0.972122 q^{67} -0.891218 q^{69} +9.54106 q^{71} -13.1975 q^{73} +3.18398 q^{75} +17.9343 q^{77} +4.95916 q^{79} +8.56079 q^{81} +6.33684 q^{83} +5.76637 q^{85} -0.320757 q^{87} -9.69747 q^{89} +1.47127 q^{91} +0.979347 q^{93} -31.2901 q^{95} -4.58016 q^{97} -10.2386 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * q^99

Introduction

L-functions

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