LMFDB - Embedded newform 6028.2.a.f.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, 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("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$0$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.58135 q^{3} +3.56375 q^{5} +0.0558785 q^{7} -0.499338 q^{9} +O(q^{10})$	$q+1.58135 q^{3} +3.56375 q^{5} +0.0558785 q^{7} -0.499338 q^{9} +1.00000 q^{11} -5.43332 q^{13} +5.63553 q^{15} +4.07661 q^{17} +4.25952 q^{19} +0.0883634 q^{21} +1.40943 q^{23} +7.70032 q^{25} -5.53367 q^{27} +8.75513 q^{29} +1.06639 q^{31} +1.58135 q^{33} +0.199137 q^{35} +8.52262 q^{37} -8.59198 q^{39} -8.89755 q^{41} -4.21495 q^{43} -1.77952 q^{45} +9.28749 q^{47} -6.99688 q^{49} +6.44653 q^{51} +1.73178 q^{53} +3.56375 q^{55} +6.73579 q^{57} +4.75618 q^{59} +11.9320 q^{61} -0.0279023 q^{63} -19.3630 q^{65} +4.87950 q^{67} +2.22881 q^{69} -3.42672 q^{71} -13.7409 q^{73} +12.1769 q^{75} +0.0558785 q^{77} +3.86395 q^{79} -7.25265 q^{81} -13.2022 q^{83} +14.5280 q^{85} +13.8449 q^{87} -8.42230 q^{89} -0.303606 q^{91} +1.68633 q^{93} +15.1799 q^{95} +16.6002 q^{97} -0.499338 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) $ 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9	$ 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9} + 29 q^{11} + 10 q^{15} + 29 q^{17} + 7 q^{19} + 2 q^{21} + 36 q^{23} + 36 q^{25} + 50 q^{27} + 9 q^{29} + 28 q^{31} + 14 q^{33} + 15 q^{35} + 25 q^{37} + 9 q^{39} + 19 q^{41} + 23 q^{43} + 5 q^{45} + 27 q^{47} + 27 q^{49} + 13 q^{51} + 4 q^{53} + 9 q^{55} + 14 q^{57} + 40 q^{59} + 20 q^{61} - 17 q^{63} + 9 q^{65} + 59 q^{67} + 30 q^{69} + 29 q^{71} - 5 q^{73} + 46 q^{75} + 14 q^{77} + 29 q^{79} + 61 q^{81} + 35 q^{83} - 57 q^{85} + 45 q^{87} + 39 q^{89} + 45 q^{91} - 8 q^{93} + q^{95} + 55 q^{97} + 43 q^{99}+O(q^{100}) $ 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9 + 29 * q^11 + 10 * q^15 + 29 * q^17 + 7 * q^19 + 2 * q^21 + 36 * q^23 + 36 * q^25 + 50 * q^27 + 9 * q^29 + 28 * q^31 + 14 * q^33 + 15 * q^35 + 25 * q^37 + 9 * q^39 + 19 * q^41 + 23 * q^43 + 5 * q^45 + 27 * q^47 + 27 * q^49 + 13 * q^51 + 4 * q^53 + 9 * q^55 + 14 * q^57 + 40 * q^59 + 20 * q^61 - 17 * q^63 + 9 * q^65 + 59 * q^67 + 30 * q^69 + 29 * q^71 - 5 * q^73 + 46 * q^75 + 14 * q^77 + 29 * q^79 + 61 * q^81 + 35 * q^83 - 57 * q^85 + 45 * q^87 + 39 * q^89 + 45 * q^91 - 8 * q^93 + q^95 + 55 * q^97 + 43 * 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
$3$	1.58135	0.912992	0.456496	−	0.889726i	$-0.349104\pi$
$3$	1.58135	0.912992	0.456496	+	0.889726i	$0.349104\pi$
$4$	0	0
$5$	3.56375	1.59376	0.796879	−	0.604139i	$-0.206483\pi$
$5$	3.56375	1.59376	0.796879	+	0.604139i	$0.206483\pi$
$6$	0	0
$7$	0.0558785	0.0211201	0.0105600	−	0.999944i	$-0.496639\pi$
$7$	0.0558785	0.0211201	0.0105600	+	0.999944i	$0.496639\pi$
$8$	0	0
$9$	−0.499338	−0.166446
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.43332	−1.50693	−0.753466	−	0.657486i	$-0.771620\pi$
$13$	−5.43332	−1.50693	−0.753466	+	0.657486i	$0.771620\pi$
$14$	0	0
$15$	5.63553	1.45509
$16$	0	0
$17$	4.07661	0.988722	0.494361	−	0.869257i	$-0.335402\pi$
$17$	4.07661	0.988722	0.494361	+	0.869257i	$0.335402\pi$
$18$	0	0
$19$	4.25952	0.977202	0.488601	−	0.872507i	$-0.337507\pi$
$19$	4.25952	0.977202	0.488601	+	0.872507i	$0.337507\pi$
$20$	0	0
$21$	0.0883634	0.0192825
$22$	0	0
$23$	1.40943	0.293887	0.146944	−	0.989145i	$-0.453056\pi$
$23$	1.40943	0.293887	0.146944	+	0.989145i	$0.453056\pi$
$24$	0	0
$25$	7.70032	1.54006
$26$	0	0
$27$	−5.53367	−1.06496
$28$	0	0
$29$	8.75513	1.62579	0.812893	−	0.582413i	$-0.197891\pi$
$29$	8.75513	1.62579	0.812893	+	0.582413i	$0.197891\pi$
$30$	0	0
$31$	1.06639	0.191529	0.0957646	−	0.995404i	$-0.469470\pi$
$31$	1.06639	0.191529	0.0957646	+	0.995404i	$0.469470\pi$
$32$	0	0
$33$	1.58135	0.275277
$34$	0	0
$35$	0.199137	0.0336603
$36$	0	0
$37$	8.52262	1.40111	0.700555	−	0.713598i	$-0.252936\pi$
$37$	8.52262	1.40111	0.700555	+	0.713598i	$0.252936\pi$
$38$	0	0
$39$	−8.59198	−1.37582
$40$	0	0
$41$	−8.89755	−1.38956	−0.694782	−	0.719221i	$-0.744499\pi$
$41$	−8.89755	−1.38956	−0.694782	+	0.719221i	$0.744499\pi$
$42$	0	0
$43$	−4.21495	−0.642774	−0.321387	−	0.946948i	$-0.604149\pi$
$43$	−4.21495	−0.642774	−0.321387	+	0.946948i	$0.604149\pi$
$44$	0	0
$45$	−1.77952	−0.265275
$46$	0	0
$47$	9.28749	1.35472	0.677360	−	0.735652i	$-0.263124\pi$
$47$	9.28749	1.35472	0.677360	+	0.735652i	$0.263124\pi$
$48$	0	0
$49$	−6.99688	−0.999554
$50$	0	0
$51$	6.44653	0.902695
$52$	0	0
$53$	1.73178	0.237879	0.118939	−	0.992902i	$-0.462051\pi$
$53$	1.73178	0.237879	0.118939	+	0.992902i	$0.462051\pi$
$54$	0	0
$55$	3.56375	0.480536
$56$	0	0
$57$	6.73579	0.892177
$58$	0	0
$59$	4.75618	0.619201	0.309601	−	0.950867i	$-0.399805\pi$
$59$	4.75618	0.619201	0.309601	+	0.950867i	$0.399805\pi$
$60$	0	0
$61$	11.9320	1.52774	0.763869	−	0.645372i	$-0.223298\pi$
$61$	11.9320	1.52774	0.763869	+	0.645372i	$0.223298\pi$
$62$	0	0
$63$	−0.0279023	−0.00351536
$64$	0	0
$65$	−19.3630	−2.40169
$66$	0	0
$67$	4.87950	0.596126	0.298063	−	0.954546i	$-0.403659\pi$
$67$	4.87950	0.596126	0.298063	+	0.954546i	$0.403659\pi$
$68$	0	0
$69$	2.22881	0.268317
$70$	0	0
$71$	−3.42672	−0.406676	−0.203338	−	0.979109i	$-0.565179\pi$
$71$	−3.42672	−0.406676	−0.203338	+	0.979109i	$0.565179\pi$
$72$	0	0
$73$	−13.7409	−1.60825	−0.804127	−	0.594457i	$-0.797367\pi$
$73$	−13.7409	−1.60825	−0.804127	+	0.594457i	$0.797367\pi$
$74$	0	0
$75$	12.1769	1.40607
$76$	0	0
$77$	0.0558785	0.00636795
$78$	0	0
$79$	3.86395	0.434728	0.217364	−	0.976091i	$-0.430254\pi$
$79$	3.86395	0.434728	0.217364	+	0.976091i	$0.430254\pi$
$80$	0	0
$81$	−7.25265	−0.805850
$82$	0	0
$83$	−13.2022	−1.44913	−0.724566	−	0.689206i	$-0.757960\pi$
$83$	−13.2022	−1.44913	−0.724566	+	0.689206i	$0.757960\pi$
$84$	0	0
$85$	14.5280	1.57578
$86$	0	0
$87$	13.8449	1.48433
$88$	0	0
$89$	−8.42230	−0.892762	−0.446381	−	0.894843i	$-0.647287\pi$
$89$	−8.42230	−0.892762	−0.446381	+	0.894843i	$0.647287\pi$
$90$	0	0
$91$	−0.303606	−0.0318266
$92$	0	0
$93$	1.68633	0.174865
$94$	0	0
$95$	15.1799	1.55742
$96$	0	0
$97$	16.6002	1.68549	0.842746	−	0.538311i	$-0.180937\pi$
$97$	16.6002	1.68549	0.842746	+	0.538311i	$0.180937\pi$
$98$	0	0
$99$	−0.499338	−0.0501854
$100$	0	0
$101$	8.08123	0.804112	0.402056	−	0.915615i	$-0.368296\pi$
$101$	8.08123	0.804112	0.402056	+	0.915615i	$0.368296\pi$
$102$	0	0
$103$	−4.31567	−0.425235	−0.212618	−	0.977135i	$-0.568199\pi$
$103$	−4.31567	−0.425235	−0.212618	+	0.977135i	$0.568199\pi$
$104$	0	0
$105$	0.314905	0.0307316
$106$	0	0
$107$	13.0398	1.26061	0.630305	−	0.776348i	$-0.282930\pi$
$107$	13.0398	1.26061	0.630305	+	0.776348i	$0.282930\pi$
$108$	0	0
$109$	−13.0007	−1.24524	−0.622620	−	0.782524i	$-0.713932\pi$
$109$	−13.0007	−1.24524	−0.622620	+	0.782524i	$0.713932\pi$
$110$	0	0
$111$	13.4772	1.27920
$112$	0	0
$113$	14.7949	1.39179	0.695893	−	0.718145i	$-0.255009\pi$
$113$	14.7949	1.39179	0.695893	+	0.718145i	$0.255009\pi$
$114$	0	0
$115$	5.02287	0.468385
$116$	0	0
$117$	2.71307	0.250823
$118$	0	0
$119$	0.227795	0.0208819
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−14.0701	−1.26866
$124$	0	0
$125$	9.62325	0.860730
$126$	0	0
$127$	−6.94620	−0.616375	−0.308188	−	0.951326i	$-0.599722\pi$
$127$	−6.94620	−0.616375	−0.308188	+	0.951326i	$0.599722\pi$
$128$	0	0
$129$	−6.66530	−0.586847
$130$	0	0
$131$	3.68408	0.321880	0.160940	−	0.986964i	$-0.448547\pi$
$131$	3.68408	0.321880	0.160940	+	0.986964i	$0.448547\pi$
$132$	0	0
$133$	0.238016	0.0206386
$134$	0	0
$135$	−19.7206	−1.69728
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	1.31316	0.111381	0.0556903	−	0.998448i	$-0.482264\pi$
$139$	1.31316	0.111381	0.0556903	+	0.998448i	$0.482264\pi$
$140$	0	0
$141$	14.6868	1.23685
$142$	0	0
$143$	−5.43332	−0.454357
$144$	0	0
$145$	31.2011	2.59111
$146$	0	0
$147$	−11.0645	−0.912585
$148$	0	0
$149$	7.08513	0.580436	0.290218	−	0.956961i	$-0.406272\pi$
$149$	7.08513	0.580436	0.290218	+	0.956961i	$0.406272\pi$
$150$	0	0
$151$	17.5539	1.42852	0.714258	−	0.699882i	$-0.246764\pi$
$151$	17.5539	1.42852	0.714258	+	0.699882i	$0.246764\pi$
$152$	0	0
$153$	−2.03560	−0.164569
$154$	0	0
$155$	3.80034	0.305251
$156$	0	0
$157$	17.9365	1.43149	0.715743	−	0.698364i	$-0.246088\pi$
$157$	17.9365	1.43149	0.715743	+	0.698364i	$0.246088\pi$
$158$	0	0
$159$	2.73855	0.217181
$160$	0	0
$161$	0.0787571	0.00620693
$162$	0	0
$163$	−0.901355	−0.0705996	−0.0352998	−	0.999377i	$-0.511239\pi$
$163$	−0.901355	−0.0705996	−0.0352998	+	0.999377i	$0.511239\pi$
$164$	0	0
$165$	5.63553	0.438725
$166$	0	0
$167$	−15.6682	−1.21244	−0.606222	−	0.795295i	$-0.707316\pi$
$167$	−15.6682	−1.21244	−0.606222	+	0.795295i	$0.707316\pi$
$168$	0	0
$169$	16.5210	1.27085
$170$	0	0
$171$	−2.12694	−0.162651
$172$	0	0
$173$	−11.0814	−0.842500	−0.421250	−	0.906944i	$-0.638409\pi$
$173$	−11.0814	−0.842500	−0.421250	+	0.906944i	$0.638409\pi$
$174$	0	0
$175$	0.430282	0.0325263
$176$	0	0
$177$	7.52117	0.565326
$178$	0	0
$179$	17.5912	1.31483	0.657414	−	0.753529i	$-0.271650\pi$
$179$	17.5912	1.31483	0.657414	+	0.753529i	$0.271650\pi$
$180$	0	0
$181$	3.37543	0.250894	0.125447	−	0.992100i	$-0.459964\pi$
$181$	3.37543	0.250894	0.125447	+	0.992100i	$0.459964\pi$
$182$	0	0
$183$	18.8687	1.39481
$184$	0	0
$185$	30.3725	2.23303
$186$	0	0
$187$	4.07661	0.298111
$188$	0	0
$189$	−0.309213	−0.0224920
$190$	0	0
$191$	−22.3047	−1.61391	−0.806954	−	0.590614i	$-0.798885\pi$
$191$	−22.3047	−1.61391	−0.806954	+	0.590614i	$0.798885\pi$
$192$	0	0
$193$	−7.02865	−0.505934	−0.252967	−	0.967475i	$-0.581406\pi$
$193$	−7.02865	−0.505934	−0.252967	+	0.967475i	$0.581406\pi$
$194$	0	0
$195$	−30.6197	−2.19272
$196$	0	0
$197$	−9.05524	−0.645159	−0.322579	−	0.946542i	$-0.604550\pi$
$197$	−9.05524	−0.645159	−0.322579	+	0.946542i	$0.604550\pi$
$198$	0	0
$199$	19.4464	1.37852	0.689261	−	0.724513i	$-0.257935\pi$
$199$	19.4464	1.37852	0.689261	+	0.724513i	$0.257935\pi$
$200$	0	0
$201$	7.71619	0.544258
$202$	0	0
$203$	0.489224	0.0343368
$204$	0	0
$205$	−31.7086	−2.21463
$206$	0	0
$207$	−0.703784	−0.0489164
$208$	0	0
$209$	4.25952	0.294637
$210$	0	0
$211$	24.1293	1.66113	0.830565	−	0.556922i	$-0.188018\pi$
$211$	24.1293	1.66113	0.830565	+	0.556922i	$0.188018\pi$
$212$	0	0
$213$	−5.41883	−0.371292
$214$	0	0
$215$	−15.0210	−1.02443
$216$	0	0
$217$	0.0595883	0.00404511
$218$	0	0
$219$	−21.7292	−1.46832
$220$	0	0
$221$	−22.1495	−1.48994
$222$	0	0
$223$	−13.8991	−0.930753	−0.465376	−	0.885113i	$-0.654081\pi$
$223$	−13.8991	−0.930753	−0.465376	+	0.885113i	$0.654081\pi$
$224$	0	0
$225$	−3.84506	−0.256337
$226$	0	0
$227$	−16.1178	−1.06978	−0.534888	−	0.844923i	$-0.679646\pi$
$227$	−16.1178	−1.06978	−0.534888	+	0.844923i	$0.679646\pi$
$228$	0	0
$229$	−4.12894	−0.272848	−0.136424	−	0.990651i	$-0.543561\pi$
$229$	−4.12894	−0.272848	−0.136424	+	0.990651i	$0.543561\pi$
$230$	0	0
$231$	0.0883634	0.00581389
$232$	0	0
$233$	−5.32041	−0.348552	−0.174276	−	0.984697i	$-0.555758\pi$
$233$	−5.32041	−0.348552	−0.174276	+	0.984697i	$0.555758\pi$
$234$	0	0
$235$	33.0983	2.15909
$236$	0	0
$237$	6.11025	0.396903
$238$	0	0
$239$	−8.88223	−0.574543	−0.287272	−	0.957849i	$-0.592748\pi$
$239$	−8.88223	−0.574543	−0.287272	+	0.957849i	$0.592748\pi$
$240$	0	0
$241$	−13.1929	−0.849828	−0.424914	−	0.905234i	$-0.639696\pi$
$241$	−13.1929	−0.849828	−0.424914	+	0.905234i	$0.639696\pi$
$242$	0	0
$243$	5.13205	0.329221
$244$	0	0
$245$	−24.9351	−1.59305
$246$	0	0
$247$	−23.1434	−1.47258
$248$	0	0
$249$	−20.8773	−1.32305
$250$	0	0
$251$	7.14712	0.451122	0.225561	−	0.974229i	$-0.427578\pi$
$251$	7.14712	0.451122	0.225561	+	0.974229i	$0.427578\pi$
$252$	0	0
$253$	1.40943	0.0886104
$254$	0	0
$255$	22.9738	1.43868
$256$	0	0
$257$	−8.27247	−0.516023	−0.258011	−	0.966142i	$-0.583067\pi$
$257$	−8.27247	−0.516023	−0.258011	+	0.966142i	$0.583067\pi$
$258$	0	0
$259$	0.476232	0.0295916
$260$	0	0
$261$	−4.37177	−0.270606
$262$	0	0
$263$	27.6871	1.70726	0.853628	−	0.520883i	$-0.174397\pi$
$263$	27.6871	1.70726	0.853628	+	0.520883i	$0.174397\pi$
$264$	0	0
$265$	6.17165	0.379121
$266$	0	0
$267$	−13.3186	−0.815084
$268$	0	0
$269$	−28.1649	−1.71724	−0.858622	−	0.512609i	$-0.828679\pi$
$269$	−28.1649	−1.71724	−0.858622	+	0.512609i	$0.828679\pi$
$270$	0	0
$271$	−20.9983	−1.27555	−0.637777	−	0.770221i	$-0.720146\pi$
$271$	−20.9983	−1.27555	−0.637777	+	0.770221i	$0.720146\pi$
$272$	0	0
$273$	−0.480107	−0.0290574
$274$	0	0
$275$	7.70032	0.464347
$276$	0	0
$277$	30.3273	1.82219	0.911096	−	0.412194i	$-0.135237\pi$
$277$	30.3273	1.82219	0.911096	+	0.412194i	$0.135237\pi$
$278$	0	0
$279$	−0.532489	−0.0318793
$280$	0	0
$281$	−29.6655	−1.76970	−0.884849	−	0.465878i	$-0.845739\pi$
$281$	−29.6655	−1.76970	−0.884849	+	0.465878i	$0.845739\pi$
$282$	0	0
$283$	−6.44292	−0.382992	−0.191496	−	0.981493i	$-0.561334\pi$
$283$	−6.44292	−0.382992	−0.191496	+	0.981493i	$0.561334\pi$
$284$	0	0
$285$	24.0047	1.42191
$286$	0	0
$287$	−0.497182	−0.0293477
$288$	0	0
$289$	−0.381285	−0.0224286
$290$	0	0
$291$	26.2507	1.53884
$292$	0	0
$293$	2.43454	0.142228	0.0711138	−	0.997468i	$-0.477345\pi$
$293$	2.43454	0.142228	0.0711138	+	0.997468i	$0.477345\pi$
$294$	0	0
$295$	16.9498	0.986857
$296$	0	0
$297$	−5.53367	−0.321096
$298$	0	0
$299$	−7.65791	−0.442869
$300$	0	0
$301$	−0.235525	−0.0135754
$302$	0	0
$303$	12.7792	0.734148
$304$	0	0
$305$	42.5227	2.43484
$306$	0	0
$307$	19.3877	1.10652	0.553258	−	0.833010i	$-0.313385\pi$
$307$	19.3877	1.10652	0.553258	+	0.833010i	$0.313385\pi$
$308$	0	0
$309$	−6.82457	−0.388236
$310$	0	0
$311$	−9.94812	−0.564106	−0.282053	−	0.959399i	$-0.591015\pi$
$311$	−9.94812	−0.564106	−0.282053	+	0.959399i	$0.591015\pi$
$312$	0	0
$313$	−2.01286	−0.113774	−0.0568869	−	0.998381i	$-0.518117\pi$
$313$	−2.01286	−0.113774	−0.0568869	+	0.998381i	$0.518117\pi$
$314$	0	0
$315$	−0.0994367	−0.00560263
$316$	0	0
$317$	−33.9087	−1.90450	−0.952252	−	0.305313i	$-0.901239\pi$
$317$	−33.9087	−1.90450	−0.952252	+	0.305313i	$0.901239\pi$
$318$	0	0
$319$	8.75513	0.490193
$320$	0	0
$321$	20.6205	1.15093
$322$	0	0
$323$	17.3644	0.966181
$324$	0	0
$325$	−41.8383	−2.32077
$326$	0	0
$327$	−20.5586	−1.13689
$328$	0	0
$329$	0.518971	0.0286118
$330$	0	0
$331$	−32.9506	−1.81113	−0.905565	−	0.424207i	$-0.860553\pi$
$331$	−32.9506	−1.81113	−0.905565	+	0.424207i	$0.860553\pi$
$332$	0	0
$333$	−4.25567	−0.233209
$334$	0	0
$335$	17.3893	0.950080
$336$	0	0
$337$	1.39174	0.0758127	0.0379064	−	0.999281i	$-0.487931\pi$
$337$	1.39174	0.0758127	0.0379064	+	0.999281i	$0.487931\pi$
$338$	0	0
$339$	23.3959	1.27069
$340$	0	0
$341$	1.06639	0.0577482
$342$	0	0
$343$	−0.782125	−0.0422308
$344$	0	0
$345$	7.94291	0.427632
$346$	0	0
$347$	−13.7331	−0.737231	−0.368616	−	0.929582i	$-0.620168\pi$
$347$	−13.7331	−0.737231	−0.368616	+	0.929582i	$0.620168\pi$
$348$	0	0
$349$	2.73845	0.146586	0.0732929	−	0.997310i	$-0.476649\pi$
$349$	2.73845	0.146586	0.0732929	+	0.997310i	$0.476649\pi$
$350$	0	0
$351$	30.0662	1.60482
$352$	0	0
$353$	32.8517	1.74852	0.874260	−	0.485457i	$-0.161347\pi$
$353$	32.8517	1.74852	0.874260	+	0.485457i	$0.161347\pi$
$354$	0	0
$355$	−12.2120	−0.648143
$356$	0	0
$357$	0.360223	0.0190650
$358$	0	0
$359$	−24.4443	−1.29012	−0.645061	−	0.764131i	$-0.723168\pi$
$359$	−24.4443	−1.29012	−0.645061	+	0.764131i	$0.723168\pi$
$360$	0	0
$361$	−0.856453	−0.0450765
$362$	0	0
$363$	1.58135	0.0829993
$364$	0	0
$365$	−48.9692	−2.56317
$366$	0	0
$367$	32.6267	1.70310	0.851550	−	0.524273i	$-0.175663\pi$
$367$	32.6267	1.70310	0.851550	+	0.524273i	$0.175663\pi$
$368$	0	0
$369$	4.44288	0.231287
$370$	0	0
$371$	0.0967695	0.00502402
$372$	0	0
$373$	−24.4016	−1.26347	−0.631734	−	0.775185i	$-0.717657\pi$
$373$	−24.4016	−1.26347	−0.631734	+	0.775185i	$0.717657\pi$
$374$	0	0
$375$	15.2177	0.785839
$376$	0	0
$377$	−47.5694	−2.44995
$378$	0	0
$379$	5.34121	0.274360	0.137180	−	0.990546i	$-0.456196\pi$
$379$	5.34121	0.274360	0.137180	+	0.990546i	$0.456196\pi$
$380$	0	0
$381$	−10.9844	−0.562745
$382$	0	0
$383$	17.3140	0.884702	0.442351	−	0.896842i	$-0.354144\pi$
$383$	17.3140	0.884702	0.442351	+	0.896842i	$0.354144\pi$
$384$	0	0
$385$	0.199137	0.0101490
$386$	0	0
$387$	2.10468	0.106987
$388$	0	0
$389$	−28.8752	−1.46403	−0.732016	−	0.681288i	$-0.761420\pi$
$389$	−28.8752	−1.46403	−0.732016	+	0.681288i	$0.761420\pi$
$390$	0	0
$391$	5.74571	0.290573
$392$	0	0
$393$	5.82581	0.293873
$394$	0	0
$395$	13.7702	0.692852
$396$	0	0
$397$	−21.7890	−1.09356	−0.546779	−	0.837277i	$-0.684146\pi$
$397$	−21.7890	−1.09356	−0.546779	+	0.837277i	$0.684146\pi$
$398$	0	0
$399$	0.376386	0.0188429
$400$	0	0
$401$	30.8824	1.54219	0.771097	−	0.636718i	$-0.219708\pi$
$401$	30.8824	1.54219	0.771097	+	0.636718i	$0.219708\pi$
$402$	0	0
$403$	−5.79404	−0.288622
$404$	0	0
$405$	−25.8466	−1.28433
$406$	0	0
$407$	8.52262	0.422451
$408$	0	0
$409$	−16.7412	−0.827801	−0.413900	−	0.910322i	$-0.635834\pi$
$409$	−16.7412	−0.827801	−0.413900	+	0.910322i	$0.635834\pi$
$410$	0	0
$411$	1.58135	0.0780022
$412$	0	0
$413$	0.265768	0.0130776
$414$	0	0
$415$	−47.0494	−2.30956
$416$	0	0
$417$	2.07656	0.101690
$418$	0	0
$419$	−1.89158	−0.0924098	−0.0462049	−	0.998932i	$-0.514713\pi$
$419$	−1.89158	−0.0924098	−0.0462049	+	0.998932i	$0.514713\pi$
$420$	0	0
$421$	14.6504	0.714017	0.357009	−	0.934101i	$-0.383797\pi$
$421$	14.6504	0.714017	0.357009	+	0.934101i	$0.383797\pi$
$422$	0	0
$423$	−4.63760	−0.225488
$424$	0	0
$425$	31.3912	1.52269
$426$	0	0
$427$	0.666743	0.0322660
$428$	0	0
$429$	−8.59198	−0.414825
$430$	0	0
$431$	−17.0005	−0.818887	−0.409443	−	0.912335i	$-0.634277\pi$
$431$	−17.0005	−0.818887	−0.409443	+	0.912335i	$0.634277\pi$
$432$	0	0
$433$	−10.0187	−0.481467	−0.240734	−	0.970591i	$-0.577388\pi$
$433$	−10.0187	−0.481467	−0.240734	+	0.970591i	$0.577388\pi$
$434$	0	0
$435$	49.3398	2.36566
$436$	0	0
$437$	6.00352	0.287187
$438$	0	0
$439$	28.4639	1.35851	0.679254	−	0.733904i	$-0.262304\pi$
$439$	28.4639	1.35851	0.679254	+	0.733904i	$0.262304\pi$
$440$	0	0
$441$	3.49381	0.166372
$442$	0	0
$443$	4.03302	0.191615	0.0958074	−	0.995400i	$-0.469457\pi$
$443$	4.03302	0.191615	0.0958074	+	0.995400i	$0.469457\pi$
$444$	0	0
$445$	−30.0150	−1.42285
$446$	0	0
$447$	11.2041	0.529933
$448$	0	0
$449$	−17.9150	−0.845461	−0.422730	−	0.906255i	$-0.638928\pi$
$449$	−17.9150	−0.845461	−0.422730	+	0.906255i	$0.638928\pi$
$450$	0	0
$451$	−8.89755	−0.418969
$452$	0	0
$453$	27.7588	1.30422
$454$	0	0
$455$	−1.08198	−0.0507238
$456$	0	0
$457$	−28.2168	−1.31992	−0.659962	−	0.751299i	$-0.729428\pi$
$457$	−28.2168	−1.31992	−0.659962	+	0.751299i	$0.729428\pi$
$458$	0	0
$459$	−22.5586	−1.05295
$460$	0	0
$461$	−17.0023	−0.791878	−0.395939	−	0.918277i	$-0.629581\pi$
$461$	−17.0023	−0.791878	−0.395939	+	0.918277i	$0.629581\pi$
$462$	0	0
$463$	4.38439	0.203760	0.101880	−	0.994797i	$-0.467514\pi$
$463$	4.38439	0.203760	0.101880	+	0.994797i	$0.467514\pi$
$464$	0	0
$465$	6.00967	0.278692
$466$	0	0
$467$	21.0118	0.972309	0.486155	−	0.873873i	$-0.338399\pi$
$467$	21.0118	0.972309	0.486155	+	0.873873i	$0.338399\pi$
$468$	0	0
$469$	0.272659	0.0125902
$470$	0	0
$471$	28.3638	1.30693
$472$	0	0
$473$	−4.21495	−0.193804
$474$	0	0
$475$	32.7997	1.50495
$476$	0	0
$477$	−0.864746	−0.0395940
$478$	0	0
$479$	26.0734	1.19132	0.595662	−	0.803235i	$-0.296890\pi$
$479$	26.0734	1.19132	0.595662	+	0.803235i	$0.296890\pi$
$480$	0	0
$481$	−46.3062	−2.11138
$482$	0	0
$483$	0.124542	0.00566688
$484$	0	0
$485$	59.1589	2.68627
$486$	0	0
$487$	6.01400	0.272521	0.136260	−	0.990673i	$-0.456492\pi$
$487$	6.01400	0.272521	0.136260	+	0.990673i	$0.456492\pi$
$488$	0	0
$489$	−1.42536	−0.0644568
$490$	0	0
$491$	−31.7489	−1.43281	−0.716403	−	0.697686i	$-0.754213\pi$
$491$	−31.7489	−1.43281	−0.716403	+	0.697686i	$0.754213\pi$
$492$	0	0
$493$	35.6912	1.60745
$494$	0	0
$495$	−1.77952	−0.0799833
$496$	0	0
$497$	−0.191480	−0.00858904
$498$	0	0
$499$	−4.16941	−0.186648	−0.0933242	−	0.995636i	$-0.529749\pi$
$499$	−4.16941	−0.186648	−0.0933242	+	0.995636i	$0.529749\pi$
$500$	0	0
$501$	−24.7769	−1.10695
$502$	0	0
$503$	−37.9615	−1.69262	−0.846310	−	0.532690i	$-0.821181\pi$
$503$	−37.9615	−1.69262	−0.846310	+	0.532690i	$0.821181\pi$
$504$	0	0
$505$	28.7995	1.28156
$506$	0	0
$507$	26.1255	1.16027
$508$	0	0
$509$	−15.5704	−0.690147	−0.345073	−	0.938576i	$-0.612146\pi$
$509$	−15.5704	−0.690147	−0.345073	+	0.938576i	$0.612146\pi$
$510$	0	0
$511$	−0.767823	−0.0339665
$512$	0	0
$513$	−23.5708	−1.04068
$514$	0	0
$515$	−15.3800	−0.677722
$516$	0	0
$517$	9.28749	0.408463
$518$	0	0
$519$	−17.5235	−0.769196
$520$	0	0
$521$	26.6370	1.16699	0.583493	−	0.812118i	$-0.301686\pi$
$521$	26.6370	1.16699	0.583493	+	0.812118i	$0.301686\pi$
$522$	0	0
$523$	30.9156	1.35184	0.675921	−	0.736974i	$-0.263746\pi$
$523$	30.9156	1.35184	0.675921	+	0.736974i	$0.263746\pi$
$524$	0	0
$525$	0.680426	0.0296962
$526$	0	0
$527$	4.34725	0.189369
$528$	0	0
$529$	−21.0135	−0.913630
$530$	0	0
$531$	−2.37494	−0.103064
$532$	0	0
$533$	48.3433	2.09398
$534$	0	0
$535$	46.4708	2.00911
$536$	0	0
$537$	27.8178	1.20043
$538$	0	0
$539$	−6.99688	−0.301377
$540$	0	0
$541$	−19.4955	−0.838175	−0.419088	−	0.907946i	$-0.637650\pi$
$541$	−19.4955	−0.838175	−0.419088	+	0.907946i	$0.637650\pi$
$542$	0	0
$543$	5.33773	0.229064
$544$	0	0
$545$	−46.3312	−1.98461
$546$	0	0
$547$	−6.45134	−0.275839	−0.137920	−	0.990443i	$-0.544042\pi$
$547$	−6.45134	−0.275839	−0.137920	+	0.990443i	$0.544042\pi$
$548$	0	0
$549$	−5.95811	−0.254286
$550$	0	0
$551$	37.2927	1.58872
$552$	0	0
$553$	0.215912	0.00918150
$554$	0	0
$555$	48.0295	2.03874
$556$	0	0
$557$	39.6562	1.68029	0.840143	−	0.542365i	$-0.182471\pi$
$557$	39.6562	1.68029	0.840143	+	0.542365i	$0.182471\pi$
$558$	0	0
$559$	22.9012	0.968617
$560$	0	0
$561$	6.44653	0.272173
$562$	0	0
$563$	29.2548	1.23294	0.616472	−	0.787377i	$-0.288561\pi$
$563$	29.2548	1.23294	0.616472	+	0.787377i	$0.288561\pi$
$564$	0	0
$565$	52.7253	2.21817
$566$	0	0
$567$	−0.405267	−0.0170196
$568$	0	0
$569$	29.9594	1.25596	0.627982	−	0.778228i	$-0.283881\pi$
$569$	29.9594	1.25596	0.627982	+	0.778228i	$0.283881\pi$
$570$	0	0
$571$	−5.38788	−0.225476	−0.112738	−	0.993625i	$-0.535962\pi$
$571$	−5.38788	−0.225476	−0.112738	+	0.993625i	$0.535962\pi$
$572$	0	0
$573$	−35.2714	−1.47348
$574$	0	0
$575$	10.8531	0.452605
$576$	0	0
$577$	−13.1101	−0.545780	−0.272890	−	0.962045i	$-0.587979\pi$
$577$	−13.1101	−0.545780	−0.272890	+	0.962045i	$0.587979\pi$
$578$	0	0
$579$	−11.1147	−0.461913
$580$	0	0
$581$	−0.737720	−0.0306058
$582$	0	0
$583$	1.73178	0.0717232
$584$	0	0
$585$	9.66869	0.399751
$586$	0	0
$587$	16.9677	0.700333	0.350167	−	0.936687i	$-0.386125\pi$
$587$	16.9677	0.700333	0.350167	+	0.936687i	$0.386125\pi$
$588$	0	0
$589$	4.54231	0.187163
$590$	0	0
$591$	−14.3195	−0.589025
$592$	0	0
$593$	−7.45809	−0.306267	−0.153133	−	0.988206i	$-0.548936\pi$
$593$	−7.45809	−0.306267	−0.153133	+	0.988206i	$0.548936\pi$
$594$	0	0
$595$	0.811803	0.0332807
$596$	0	0
$597$	30.7516	1.25858
$598$	0	0
$599$	16.8900	0.690108	0.345054	−	0.938583i	$-0.387861\pi$
$599$	16.8900	0.690108	0.345054	+	0.938583i	$0.387861\pi$
$600$	0	0
$601$	−26.1209	−1.06550	−0.532748	−	0.846274i	$-0.678841\pi$
$601$	−26.1209	−1.06550	−0.532748	+	0.846274i	$0.678841\pi$
$602$	0	0
$603$	−2.43652	−0.0992228
$604$	0	0
$605$	3.56375	0.144887
$606$	0	0
$607$	−16.7290	−0.679009	−0.339504	−	0.940604i	$-0.610259\pi$
$607$	−16.7290	−0.679009	−0.339504	+	0.940604i	$0.610259\pi$
$608$	0	0
$609$	0.773633	0.0313492
$610$	0	0
$611$	−50.4619	−2.04147
$612$	0	0
$613$	−11.5728	−0.467419	−0.233710	−	0.972306i	$-0.575087\pi$
$613$	−11.5728	−0.467419	−0.233710	+	0.972306i	$0.575087\pi$
$614$	0	0
$615$	−50.1424	−2.02194
$616$	0	0
$617$	−29.9996	−1.20774	−0.603868	−	0.797084i	$-0.706375\pi$
$617$	−29.9996	−1.20774	−0.603868	+	0.797084i	$0.706375\pi$
$618$	0	0
$619$	23.2247	0.933479	0.466739	−	0.884395i	$-0.345429\pi$
$619$	23.2247	0.933479	0.466739	+	0.884395i	$0.345429\pi$
$620$	0	0
$621$	−7.79935	−0.312977
$622$	0	0
$623$	−0.470626	−0.0188552
$624$	0	0
$625$	−4.20671	−0.168269
$626$	0	0
$627$	6.73579	0.269002
$628$	0	0
$629$	34.7434	1.38531
$630$	0	0
$631$	−34.9727	−1.39224	−0.696121	−	0.717925i	$-0.745092\pi$
$631$	−34.9727	−1.39224	−0.696121	+	0.717925i	$0.745092\pi$
$632$	0	0
$633$	38.1568	1.51660
$634$	0	0
$635$	−24.7545	−0.982353
$636$	0	0
$637$	38.0163	1.50626
$638$	0	0
$639$	1.71109	0.0676896
$640$	0	0
$641$	14.0211	0.553801	0.276901	−	0.960899i	$-0.410693\pi$
$641$	14.0211	0.553801	0.276901	+	0.960899i	$0.410693\pi$
$642$	0	0
$643$	19.1233	0.754149	0.377074	−	0.926183i	$-0.376930\pi$
$643$	19.1233	0.754149	0.377074	+	0.926183i	$0.376930\pi$
$644$	0	0
$645$	−23.7535	−0.935292
$646$	0	0
$647$	−39.1960	−1.54095	−0.770476	−	0.637469i	$-0.779981\pi$
$647$	−39.1960	−1.54095	−0.770476	+	0.637469i	$0.779981\pi$
$648$	0	0
$649$	4.75618	0.186696
$650$	0	0
$651$	0.0942298	0.00369316
$652$	0	0
$653$	2.12665	0.0832223	0.0416112	−	0.999134i	$-0.486751\pi$
$653$	2.12665	0.0832223	0.0416112	+	0.999134i	$0.486751\pi$
$654$	0	0
$655$	13.1291	0.512998
$656$	0	0
$657$	6.86137	0.267688
$658$	0	0
$659$	−25.3542	−0.987659	−0.493829	−	0.869559i	$-0.664403\pi$
$659$	−25.3542	−0.987659	−0.493829	+	0.869559i	$0.664403\pi$
$660$	0	0
$661$	21.0993	0.820668	0.410334	−	0.911935i	$-0.365412\pi$
$661$	21.0993	0.820668	0.410334	+	0.911935i	$0.365412\pi$
$662$	0	0
$663$	−35.0261	−1.36030
$664$	0	0
$665$	0.848229	0.0328929
$666$	0	0
$667$	12.3398	0.477798
$668$	0	0
$669$	−21.9793	−0.849770
$670$	0	0
$671$	11.9320	0.460630
$672$	0	0
$673$	−12.4555	−0.480124	−0.240062	−	0.970758i	$-0.577168\pi$
$673$	−12.4555	−0.480124	−0.240062	+	0.970758i	$0.577168\pi$
$674$	0	0
$675$	−42.6110	−1.64010
$676$	0	0
$677$	−11.4599	−0.440442	−0.220221	−	0.975450i	$-0.570678\pi$
$677$	−11.4599	−0.440442	−0.220221	+	0.975450i	$0.570678\pi$
$678$	0	0
$679$	0.927593	0.0355978
$680$	0	0
$681$	−25.4879	−0.976698
$682$	0	0
$683$	21.7250	0.831282	0.415641	−	0.909529i	$-0.363557\pi$
$683$	21.7250	0.831282	0.415641	+	0.909529i	$0.363557\pi$
$684$	0	0
$685$	3.56375	0.136164
$686$	0	0
$687$	−6.52928	−0.249108
$688$	0	0
$689$	−9.40934	−0.358467
$690$	0	0
$691$	27.4093	1.04270	0.521350	−	0.853343i	$-0.325429\pi$
$691$	27.4093	1.04270	0.521350	+	0.853343i	$0.325429\pi$
$692$	0	0
$693$	−0.0279023	−0.00105992
$694$	0	0
$695$	4.67977	0.177514
$696$	0	0
$697$	−36.2718	−1.37389
$698$	0	0
$699$	−8.41343	−0.318225
$700$	0	0
$701$	−50.7015	−1.91497	−0.957484	−	0.288485i	$-0.906849\pi$
$701$	−50.7015	−1.91497	−0.957484	+	0.288485i	$0.906849\pi$
$702$	0	0
$703$	36.3023	1.36917
$704$	0	0
$705$	52.3399	1.97124
$706$	0	0
$707$	0.451567	0.0169829
$708$	0	0
$709$	34.8414	1.30850	0.654248	−	0.756280i	$-0.272985\pi$
$709$	34.8414	1.30850	0.654248	+	0.756280i	$0.272985\pi$
$710$	0	0
$711$	−1.92942	−0.0723588
$712$	0	0
$713$	1.50301	0.0562880
$714$	0	0
$715$	−19.3630	−0.724136
$716$	0	0
$717$	−14.0459	−0.524553
$718$	0	0
$719$	−49.2547	−1.83689	−0.918445	−	0.395549i	$-0.870554\pi$
$719$	−49.2547	−1.83689	−0.918445	+	0.395549i	$0.870554\pi$
$720$	0	0
$721$	−0.241153	−0.00898101
$722$	0	0
$723$	−20.8625	−0.775886
$724$	0	0
$725$	67.4172	2.50381
$726$	0	0
$727$	−27.2845	−1.01193	−0.505963	−	0.862555i	$-0.668863\pi$
$727$	−27.2845	−1.01193	−0.505963	+	0.862555i	$0.668863\pi$
$728$	0	0
$729$	29.8735	1.10643
$730$	0	0
$731$	−17.1827	−0.635525
$732$	0	0
$733$	−5.12758	−0.189391	−0.0946957	−	0.995506i	$-0.530188\pi$
$733$	−5.12758	−0.189391	−0.0946957	+	0.995506i	$0.530188\pi$
$734$	0	0
$735$	−39.4311	−1.45444
$736$	0	0
$737$	4.87950	0.179739
$738$	0	0
$739$	6.95763	0.255941	0.127970	−	0.991778i	$-0.459154\pi$
$739$	6.95763	0.255941	0.127970	+	0.991778i	$0.459154\pi$
$740$	0	0
$741$	−36.5977	−1.34445
$742$	0	0
$743$	−25.2357	−0.925806	−0.462903	−	0.886409i	$-0.653192\pi$
$743$	−25.2357	−0.925806	−0.462903	+	0.886409i	$0.653192\pi$
$744$	0	0
$745$	25.2496	0.925075
$746$	0	0
$747$	6.59237	0.241202
$748$	0	0
$749$	0.728648	0.0266242
$750$	0	0
$751$	21.5511	0.786411	0.393206	−	0.919451i	$-0.371366\pi$
$751$	21.5511	0.786411	0.393206	+	0.919451i	$0.371366\pi$
$752$	0	0
$753$	11.3021	0.411871
$754$	0	0
$755$	62.5577	2.27671
$756$	0	0
$757$	4.12069	0.149769	0.0748846	−	0.997192i	$-0.476141\pi$
$757$	4.12069	0.149769	0.0748846	+	0.997192i	$0.476141\pi$
$758$	0	0
$759$	2.22881	0.0809005
$760$	0	0
$761$	33.2892	1.20673	0.603367	−	0.797464i	$-0.293825\pi$
$761$	33.2892	1.20673	0.603367	+	0.797464i	$0.293825\pi$
$762$	0	0
$763$	−0.726459	−0.0262996
$764$	0	0
$765$	−7.25439	−0.262283
$766$	0	0
$767$	−25.8418	−0.933095
$768$	0	0
$769$	−41.4390	−1.49433	−0.747165	−	0.664639i	$-0.768585\pi$
$769$	−41.4390	−1.49433	−0.747165	+	0.664639i	$0.768585\pi$
$770$	0	0
$771$	−13.0817	−0.471124
$772$	0	0
$773$	5.22552	0.187949	0.0939744	−	0.995575i	$-0.470043\pi$
$773$	5.22552	0.187949	0.0939744	+	0.995575i	$0.470043\pi$
$774$	0	0
$775$	8.21153	0.294967
$776$	0	0
$777$	0.753088	0.0270169
$778$	0	0
$779$	−37.8993	−1.35788
$780$	0	0
$781$	−3.42672	−0.122618
$782$	0	0
$783$	−48.4480	−1.73139
$784$	0	0
$785$	63.9211	2.28144
$786$	0	0
$787$	45.9799	1.63901	0.819504	−	0.573074i	$-0.194249\pi$
$787$	45.9799	1.63901	0.819504	+	0.573074i	$0.194249\pi$
$788$	0	0
$789$	43.7829	1.55871
$790$	0	0
$791$	0.826717	0.0293947
$792$	0	0
$793$	−64.8305	−2.30220
$794$	0	0
$795$	9.75952	0.346135
$796$	0	0
$797$	−5.33977	−0.189144	−0.0945722	−	0.995518i	$-0.530148\pi$
$797$	−5.33977	−0.189144	−0.0945722	+	0.995518i	$0.530148\pi$
$798$	0	0
$799$	37.8614	1.33944
$800$	0	0
$801$	4.20557	0.148597
$802$	0	0
$803$	−13.7409	−0.484907
$804$	0	0
$805$	0.280671	0.00989234
$806$	0	0
$807$	−44.5385	−1.56783
$808$	0	0
$809$	−42.9189	−1.50895	−0.754474	−	0.656330i	$-0.772108\pi$
$809$	−42.9189	−1.50895	−0.754474	+	0.656330i	$0.772108\pi$
$810$	0	0
$811$	−23.9314	−0.840345	−0.420173	−	0.907444i	$-0.638030\pi$
$811$	−23.9314	−0.840345	−0.420173	+	0.907444i	$0.638030\pi$
$812$	0	0
$813$	−33.2056	−1.16457
$814$	0	0
$815$	−3.21221	−0.112519
$816$	0	0
$817$	−17.9537	−0.628120
$818$	0	0
$819$	0.151602	0.00529741
$820$	0	0
$821$	−10.0239	−0.349836	−0.174918	−	0.984583i	$-0.555966\pi$
$821$	−10.0239	−0.349836	−0.174918	+	0.984583i	$0.555966\pi$
$822$	0	0
$823$	35.5431	1.23896	0.619478	−	0.785014i	$-0.287344\pi$
$823$	35.5431	1.23896	0.619478	+	0.785014i	$0.287344\pi$
$824$	0	0
$825$	12.1769	0.423945
$826$	0	0
$827$	9.23123	0.321001	0.160501	−	0.987036i	$-0.448689\pi$
$827$	9.23123	0.321001	0.160501	+	0.987036i	$0.448689\pi$
$828$	0	0
$829$	16.2155	0.563188	0.281594	−	0.959534i	$-0.409137\pi$
$829$	16.2155	0.563188	0.281594	+	0.959534i	$0.409137\pi$
$830$	0	0
$831$	47.9581	1.66365
$832$	0	0
$833$	−28.5235	−0.988281
$834$	0	0
$835$	−55.8377	−1.93234
$836$	0	0
$837$	−5.90105	−0.203970
$838$	0	0
$839$	34.6262	1.19543	0.597714	−	0.801709i	$-0.296076\pi$
$839$	34.6262	1.19543	0.597714	+	0.801709i	$0.296076\pi$
$840$	0	0
$841$	47.6522	1.64318
$842$	0	0
$843$	−46.9115	−1.61572
$844$	0	0
$845$	58.8768	2.02542
$846$	0	0
$847$	0.0558785	0.00192001
$848$	0	0
$849$	−10.1885	−0.349669
$850$	0	0
$851$	12.0121	0.411769
$852$	0	0
$853$	−51.5269	−1.76425	−0.882124	−	0.471017i	$-0.843887\pi$
$853$	−51.5269	−1.76425	−0.882124	+	0.471017i	$0.843887\pi$
$854$	0	0
$855$	−7.57989	−0.259227
$856$	0	0
$857$	3.07944	0.105192	0.0525958	−	0.998616i	$-0.483251\pi$
$857$	3.07944	0.105192	0.0525958	+	0.998616i	$0.483251\pi$
$858$	0	0
$859$	−13.1754	−0.449540	−0.224770	−	0.974412i	$-0.572163\pi$
$859$	−13.1754	−0.449540	−0.224770	+	0.974412i	$0.572163\pi$
$860$	0	0
$861$	−0.786218	−0.0267942
$862$	0	0
$863$	−42.4539	−1.44515	−0.722574	−	0.691293i	$-0.757041\pi$
$863$	−42.4539	−1.44515	−0.722574	+	0.691293i	$0.757041\pi$
$864$	0	0
$865$	−39.4912	−1.34274
$866$	0	0
$867$	−0.602945	−0.0204771
$868$	0	0
$869$	3.86395	0.131076
$870$	0	0
$871$	−26.5119	−0.898322
$872$	0	0
$873$	−8.28910	−0.280544
$874$	0	0
$875$	0.537733	0.0181787
$876$	0	0
$877$	−9.46215	−0.319514	−0.159757	−	0.987156i	$-0.551071\pi$
$877$	−9.46215	−0.319514	−0.159757	+	0.987156i	$0.551071\pi$
$878$	0	0
$879$	3.84986	0.129853
$880$	0	0
$881$	29.0279	0.977973	0.488987	−	0.872291i	$-0.337367\pi$
$881$	29.0279	0.977973	0.488987	+	0.872291i	$0.337367\pi$
$882$	0	0
$883$	16.8831	0.568161	0.284080	−	0.958800i	$-0.408312\pi$
$883$	16.8831	0.568161	0.284080	+	0.958800i	$0.408312\pi$
$884$	0	0
$885$	26.8036	0.900992
$886$	0	0
$887$	23.1836	0.778429	0.389214	−	0.921147i	$-0.372747\pi$
$887$	23.1836	0.778429	0.389214	+	0.921147i	$0.372747\pi$
$888$	0	0
$889$	−0.388143	−0.0130179
$890$	0	0
$891$	−7.25265	−0.242973
$892$	0	0
$893$	39.5603	1.32383
$894$	0	0
$895$	62.6907	2.09552
$896$	0	0
$897$	−12.1098	−0.404335
$898$	0	0
$899$	9.33637	0.311385
$900$	0	0
$901$	7.05980	0.235196
$902$	0	0
$903$	−0.372447	−0.0123943
$904$	0	0
$905$	12.0292	0.399864
$906$	0	0
$907$	49.4744	1.64277	0.821385	−	0.570374i	$-0.193202\pi$
$907$	49.4744	1.64277	0.821385	+	0.570374i	$0.193202\pi$
$908$	0	0
$909$	−4.03527	−0.133841
$910$	0	0
$911$	−48.5820	−1.60959	−0.804796	−	0.593551i	$-0.797726\pi$
$911$	−48.5820	−1.60959	−0.804796	+	0.593551i	$0.797726\pi$
$912$	0	0
$913$	−13.2022	−0.436930
$914$	0	0
$915$	67.2432	2.22299
$916$	0	0
$917$	0.205861	0.00679813
$918$	0	0
$919$	−27.6868	−0.913303	−0.456652	−	0.889646i	$-0.650951\pi$
$919$	−27.6868	−0.913303	−0.456652	+	0.889646i	$0.650951\pi$
$920$	0	0
$921$	30.6587	1.01024
$922$	0	0
$923$	18.6185	0.612834
$924$	0	0
$925$	65.6269	2.15780
$926$	0	0
$927$	2.15498	0.0707787
$928$	0	0
$929$	9.55188	0.313387	0.156694	−	0.987647i	$-0.449917\pi$
$929$	9.55188	0.313387	0.156694	+	0.987647i	$0.449917\pi$
$930$	0	0
$931$	−29.8034	−0.976766
$932$	0	0
$933$	−15.7314	−0.515024
$934$	0	0
$935$	14.5280	0.475117
$936$	0	0
$937$	−19.6539	−0.642064	−0.321032	−	0.947068i	$-0.604030\pi$
$937$	−19.6539	−0.642064	−0.321032	+	0.947068i	$0.604030\pi$
$938$	0	0
$939$	−3.18304	−0.103874
$940$	0	0
$941$	−28.8259	−0.939697	−0.469848	−	0.882747i	$-0.655691\pi$
$941$	−28.8259	−0.939697	−0.469848	+	0.882747i	$0.655691\pi$
$942$	0	0
$943$	−12.5405	−0.408375
$944$	0	0
$945$	−1.10196	−0.0358467
$946$	0	0
$947$	−36.3473	−1.18113	−0.590564	−	0.806991i	$-0.701095\pi$
$947$	−36.3473	−1.18113	−0.590564	+	0.806991i	$0.701095\pi$
$948$	0	0
$949$	74.6589	2.42353
$950$	0	0
$951$	−53.6215	−1.73880
$952$	0	0
$953$	14.9791	0.485220	0.242610	−	0.970124i	$-0.421997\pi$
$953$	14.9791	0.485220	0.242610	+	0.970124i	$0.421997\pi$
$954$	0	0
$955$	−79.4882	−2.57218
$956$	0	0
$957$	13.8449	0.447542
$958$	0	0
$959$	0.0558785	0.00180441
$960$	0	0
$961$	−29.8628	−0.963317
$962$	0	0
$963$	−6.51129	−0.209823
$964$	0	0
$965$	−25.0484	−0.806335
$966$	0	0
$967$	2.50066	0.0804158	0.0402079	−	0.999191i	$-0.487198\pi$
$967$	2.50066	0.0804158	0.0402079	+	0.999191i	$0.487198\pi$
$968$	0	0
$969$	27.4592	0.882115
$970$	0	0
$971$	57.4067	1.84227	0.921134	−	0.389246i	$-0.127264\pi$
$971$	57.4067	1.84227	0.921134	+	0.389246i	$0.127264\pi$
$972$	0	0
$973$	0.0733773	0.00235237
$974$	0	0
$975$	−66.1609	−2.11885
$976$	0	0
$977$	17.3093	0.553774	0.276887	−	0.960903i	$-0.410697\pi$
$977$	17.3093	0.553774	0.276887	+	0.960903i	$0.410697\pi$
$978$	0	0
$979$	−8.42230	−0.269178
$980$	0	0
$981$	6.49174	0.207265
$982$	0	0
$983$	−6.33820	−0.202157	−0.101079	−	0.994878i	$-0.532229\pi$
$983$	−6.33820	−0.202157	−0.101079	+	0.994878i	$0.532229\pi$
$984$	0	0
$985$	−32.2706	−1.02823
$986$	0	0
$987$	0.820674	0.0261223
$988$	0	0
$989$	−5.94069	−0.188903
$990$	0	0
$991$	−25.4018	−0.806916	−0.403458	−	0.914998i	$-0.632192\pi$
$991$	−25.4018	−0.806916	−0.403458	+	0.914998i	$0.632192\pi$
$992$	0	0
$993$	−52.1064	−1.65355
$994$	0	0
$995$	69.3023	2.19703
$996$	0	0
$997$	10.1593	0.321748	0.160874	−	0.986975i	$-0.448569\pi$
$997$	10.1593	0.321748	0.160874	+	0.986975i	$0.448569\pi$
$998$	0	0
$999$	−47.1614	−1.49212

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.f.1.18	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.f.1.18	✓	29	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:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q+1.58135 q^{3} +3.56375 q^{5} +0.0558785 q^{7} -0.499338 q^{9} +O(q^{10})\)	\(q+1.58135 q^{3} +3.56375 q^{5} +0.0558785 q^{7} -0.499338 q^{9} +1.00000 q^{11} -5.43332 q^{13} +5.63553 q^{15} +4.07661 q^{17} +4.25952 q^{19} +0.0883634 q^{21} +1.40943 q^{23} +7.70032 q^{25} -5.53367 q^{27} +8.75513 q^{29} +1.06639 q^{31} +1.58135 q^{33} +0.199137 q^{35} +8.52262 q^{37} -8.59198 q^{39} -8.89755 q^{41} -4.21495 q^{43} -1.77952 q^{45} +9.28749 q^{47} -6.99688 q^{49} +6.44653 q^{51} +1.73178 q^{53} +3.56375 q^{55} +6.73579 q^{57} +4.75618 q^{59} +11.9320 q^{61} -0.0279023 q^{63} -19.3630 q^{65} +4.87950 q^{67} +2.22881 q^{69} -3.42672 q^{71} -13.7409 q^{73} +12.1769 q^{75} +0.0558785 q^{77} +3.86395 q^{79} -7.25265 q^{81} -13.2022 q^{83} +14.5280 q^{85} +13.8449 q^{87} -8.42230 q^{89} -0.303606 q^{91} +1.68633 q^{93} +15.1799 q^{95} +16.6002 q^{97} -0.499338 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) \) 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9} + 29 q^{11} + 10 q^{15} + 29 q^{17} + 7 q^{19} + 2 q^{21} + 36 q^{23} + 36 q^{25} + 50 q^{27} + 9 q^{29} + 28 q^{31} + 14 q^{33} + 15 q^{35} + 25 q^{37} + 9 q^{39} + 19 q^{41} + 23 q^{43} + 5 q^{45} + 27 q^{47} + 27 q^{49} + 13 q^{51} + 4 q^{53} + 9 q^{55} + 14 q^{57} + 40 q^{59} + 20 q^{61} - 17 q^{63} + 9 q^{65} + 59 q^{67} + 30 q^{69} + 29 q^{71} - 5 q^{73} + 46 q^{75} + 14 q^{77} + 29 q^{79} + 61 q^{81} + 35 q^{83} - 57 q^{85} + 45 q^{87} + 39 q^{89} + 45 q^{91} - 8 q^{93} + q^{95} + 55 q^{97} + 43 q^{99}+O(q^{100}) \) 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9 + 29 * q^11 + 10 * q^15 + 29 * q^17 + 7 * q^19 + 2 * q^21 + 36 * q^23 + 36 * q^25 + 50 * q^27 + 9 * q^29 + 28 * q^31 + 14 * q^33 + 15 * q^35 + 25 * q^37 + 9 * q^39 + 19 * q^41 + 23 * q^43 + 5 * q^45 + 27 * q^47 + 27 * q^49 + 13 * q^51 + 4 * q^53 + 9 * q^55 + 14 * q^57 + 40 * q^59 + 20 * q^61 - 17 * q^63 + 9 * q^65 + 59 * q^67 + 30 * q^69 + 29 * q^71 - 5 * q^73 + 46 * q^75 + 14 * q^77 + 29 * q^79 + 61 * q^81 + 35 * q^83 - 57 * q^85 + 45 * q^87 + 39 * q^89 + 45 * q^91 - 8 * q^93 + q^95 + 55 * q^97 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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