LMFDB - Embedded newform 6039.2.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, 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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	no (minimal twist has level 671)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.16302 q^{2} +2.67865 q^{4} -3.42546 q^{5} +1.52570 q^{7} -1.46794 q^{8} +O(q^{10})$	\(q-2.16302 q^{2} +2.67865 q^{4} -3.42546 q^{5} +1.52570 q^{7} -1.46794 q^{8} +7.40934 q^{10} +1.00000 q^{11} +3.58739 q^{13} -3.30011 q^{14} -2.18213 q^{16} -5.94398 q^{17} -1.65639 q^{19} -9.17561 q^{20} -2.16302 q^{22} +3.88910 q^{23} +6.73378 q^{25} -7.75960 q^{26} +4.08681 q^{28} +9.39965 q^{29} -5.87279 q^{31} +7.65586 q^{32} +12.8569 q^{34} -5.22621 q^{35} +3.29618 q^{37} +3.58280 q^{38} +5.02835 q^{40} -11.2659 q^{41} +4.96645 q^{43} +2.67865 q^{44} -8.41219 q^{46} -0.704770 q^{47} -4.67225 q^{49} -14.5653 q^{50} +9.60938 q^{52} +5.18291 q^{53} -3.42546 q^{55} -2.23962 q^{56} -20.3316 q^{58} +2.75436 q^{59} +1.00000 q^{61} +12.7030 q^{62} -12.1955 q^{64} -12.2885 q^{65} +7.28569 q^{67} -15.9219 q^{68} +11.3044 q^{70} +13.0352 q^{71} +2.44588 q^{73} -7.12970 q^{74} -4.43688 q^{76} +1.52570 q^{77} -15.0610 q^{79} +7.47480 q^{80} +24.3683 q^{82} +10.3614 q^{83} +20.3609 q^{85} -10.7425 q^{86} -1.46794 q^{88} -0.417315 q^{89} +5.47327 q^{91} +10.4175 q^{92} +1.52443 q^{94} +5.67389 q^{95} -11.7699 q^{97} +10.1062 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) $ 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8	$ 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8} + q^{10} + 21 q^{11} + 20 q^{13} - 17 q^{14} + 50 q^{16} - q^{17} + 15 q^{19} + 2 q^{20} - 11 q^{23} + 48 q^{25} + 5 q^{26} - 16 q^{28} + 9 q^{29} + 22 q^{31} - 3 q^{32} + 33 q^{34} + 39 q^{35} + 21 q^{37} - 11 q^{38} - 16 q^{40} - 7 q^{41} + 16 q^{43} + 32 q^{44} - 3 q^{46} - 5 q^{47} + 80 q^{49} + 33 q^{50} + 60 q^{52} - 9 q^{53} - 7 q^{55} - 44 q^{56} - 27 q^{58} - 13 q^{59} + 21 q^{61} + 23 q^{62} + 66 q^{64} - 25 q^{65} + 38 q^{67} + 74 q^{68} - 33 q^{70} - 12 q^{71} + 20 q^{73} + 12 q^{74} + 59 q^{76} + 5 q^{77} + q^{79} + 38 q^{80} + 7 q^{82} + 19 q^{83} + 38 q^{85} + 3 q^{86} + 6 q^{88} - 37 q^{89} + 24 q^{91} - 31 q^{92} - 64 q^{94} + 43 q^{95} + 68 q^{97} + 2 q^{98}+O(q^{100}) $ 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8 + q^10 + 21 * q^11 + 20 * q^13 - 17 * q^14 + 50 * q^16 - q^17 + 15 * q^19 + 2 * q^20 - 11 * q^23 + 48 * q^25 + 5 * q^26 - 16 * q^28 + 9 * q^29 + 22 * q^31 - 3 * q^32 + 33 * q^34 + 39 * q^35 + 21 * q^37 - 11 * q^38 - 16 * q^40 - 7 * q^41 + 16 * q^43 + 32 * q^44 - 3 * q^46 - 5 * q^47 + 80 * q^49 + 33 * q^50 + 60 * q^52 - 9 * q^53 - 7 * q^55 - 44 * q^56 - 27 * q^58 - 13 * q^59 + 21 * q^61 + 23 * q^62 + 66 * q^64 - 25 * q^65 + 38 * q^67 + 74 * q^68 - 33 * q^70 - 12 * q^71 + 20 * q^73 + 12 * q^74 + 59 * q^76 + 5 * q^77 + q^79 + 38 * q^80 + 7 * q^82 + 19 * q^83 + 38 * q^85 + 3 * q^86 + 6 * q^88 - 37 * q^89 + 24 * q^91 - 31 * q^92 - 64 * q^94 + 43 * q^95 + 68 * q^97 + 2 * q^98

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 $

$2$	−2.16302	−1.52949	−0.764743	−	0.644336i	$-0.777134\pi$
$2$	−2.16302	−1.52949	−0.764743	+	0.644336i	$0.777134\pi$
$3$	0	0
$3$	0	0
$4$	2.67865	1.33933
$5$	−3.42546	−1.53191	−0.765956	−	0.642893i	$-0.777734\pi$
$5$	−3.42546	−1.53191	−0.765956	+	0.642893i	$0.777734\pi$
$6$	0	0
$7$	1.52570	0.576659	0.288329	−	0.957531i	$-0.406900\pi$
$7$	1.52570	0.576659	0.288329	+	0.957531i	$0.406900\pi$
$8$	−1.46794	−0.518994
$9$	0	0
$10$	7.40934	2.34304
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	3.58739	0.994964	0.497482	−	0.867474i	$-0.334258\pi$
$13$	3.58739	0.994964	0.497482	+	0.867474i	$0.334258\pi$
$14$	−3.30011	−0.881991
$15$	0	0
$16$	−2.18213	−0.545533
$17$	−5.94398	−1.44163	−0.720814	−	0.693129i	$-0.756232\pi$
$17$	−5.94398	−1.44163	−0.720814	+	0.693129i	$0.756232\pi$
$18$	0	0
$19$	−1.65639	−0.380001	−0.190001	−	0.981784i	$-0.560849\pi$
$19$	−1.65639	−0.380001	−0.190001	+	0.981784i	$0.560849\pi$
$20$	−9.17561	−2.05173
$21$	0	0
$22$	−2.16302	−0.461157
$23$	3.88910	0.810933	0.405467	−	0.914110i	$-0.367109\pi$
$23$	3.88910	0.810933	0.405467	+	0.914110i	$0.367109\pi$
$24$	0	0
$25$	6.73378	1.34676
$26$	−7.75960	−1.52178
$27$	0	0
$28$	4.08681	0.772334
$29$	9.39965	1.74547	0.872735	−	0.488194i	$-0.162344\pi$
$29$	9.39965	1.74547	0.872735	+	0.488194i	$0.162344\pi$
$30$	0	0
$31$	−5.87279	−1.05478	−0.527392	−	0.849622i	$-0.676830\pi$
$31$	−5.87279	−1.05478	−0.527392	+	0.849622i	$0.676830\pi$
$32$	7.65586	1.35338
$33$	0	0
$34$	12.8569	2.20495
$35$	−5.22621	−0.883391
$36$	0	0
$37$	3.29618	0.541889	0.270944	−	0.962595i	$-0.412664\pi$
$37$	3.29618	0.541889	0.270944	+	0.962595i	$0.412664\pi$
$38$	3.58280	0.581207
$39$	0	0
$40$	5.02835	0.795053
$41$	−11.2659	−1.75943	−0.879716	−	0.475500i	$-0.842267\pi$
$41$	−11.2659	−1.75943	−0.879716	+	0.475500i	$0.842267\pi$
$42$	0	0
$43$	4.96645	0.757376	0.378688	−	0.925524i	$-0.376375\pi$
$43$	4.96645	0.757376	0.378688	+	0.925524i	$0.376375\pi$
$44$	2.67865	0.403822
$45$	0	0
$46$	−8.41219	−1.24031
$47$	−0.704770	−0.102801	−0.0514007	−	0.998678i	$-0.516369\pi$
$47$	−0.704770	−0.102801	−0.0514007	+	0.998678i	$0.516369\pi$
$48$	0	0
$49$	−4.67225	−0.667464
$50$	−14.5653	−2.05984
$51$	0	0
$52$	9.60938	1.33258
$53$	5.18291	0.711927	0.355964	−	0.934500i	$-0.384153\pi$
$53$	5.18291	0.711927	0.355964	+	0.934500i	$0.384153\pi$
$54$	0	0
$55$	−3.42546	−0.461889
$56$	−2.23962	−0.299282
$57$	0	0
$58$	−20.3316	−2.66967
$59$	2.75436	0.358587	0.179294	−	0.983796i	$-0.442619\pi$
$59$	2.75436	0.358587	0.179294	+	0.983796i	$0.442619\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	12.7030	1.61328
$63$	0	0
$64$	−12.1955	−1.52444
$65$	−12.2885	−1.52420
$66$	0	0
$67$	7.28569	0.890089	0.445045	−	0.895508i	$-0.353188\pi$
$67$	7.28569	0.890089	0.445045	+	0.895508i	$0.353188\pi$
$68$	−15.9219	−1.93081
$69$	0	0
$70$	11.3044	1.35113
$71$	13.0352	1.54700	0.773498	−	0.633799i	$-0.218505\pi$
$71$	13.0352	1.54700	0.773498	+	0.633799i	$0.218505\pi$
$72$	0	0
$73$	2.44588	0.286269	0.143134	−	0.989703i	$-0.454282\pi$
$73$	2.44588	0.286269	0.143134	+	0.989703i	$0.454282\pi$
$74$	−7.12970	−0.828811
$75$	0	0
$76$	−4.43688	−0.508946
$77$	1.52570	0.173869
$78$	0	0
$79$	−15.0610	−1.69449	−0.847245	−	0.531202i	$-0.821741\pi$
$79$	−15.0610	−1.69449	−0.847245	+	0.531202i	$0.821741\pi$
$80$	7.47480	0.835708
$81$	0	0
$82$	24.3683	2.69103
$83$	10.3614	1.13731	0.568656	−	0.822575i	$-0.307463\pi$
$83$	10.3614	1.13731	0.568656	+	0.822575i	$0.307463\pi$
$84$	0	0
$85$	20.3609	2.20845
$86$	−10.7425	−1.15840
$87$	0	0
$88$	−1.46794	−0.156482
$89$	−0.417315	−0.0442353	−0.0221176	−	0.999755i	$-0.507041\pi$
$89$	−0.417315	−0.0442353	−0.0221176	+	0.999755i	$0.507041\pi$
$90$	0	0
$91$	5.47327	0.573755
$92$	10.4175	1.08610
$93$	0	0
$94$	1.52443	0.157233
$95$	5.67389	0.582129
$96$	0	0
$97$	−11.7699	−1.19505	−0.597524	−	0.801851i	$-0.703849\pi$
$97$	−11.7699	−1.19505	−0.597524	+	0.801851i	$0.703849\pi$
$98$	10.1062	1.02088
$99$	0	0
$100$	18.0374	1.80374
$101$	14.0074	1.39379	0.696895	−	0.717173i	$-0.254564\pi$
$101$	14.0074	1.39379	0.696895	+	0.717173i	$0.254564\pi$
$102$	0	0
$103$	5.51871	0.543775	0.271888	−	0.962329i	$-0.412352\pi$
$103$	5.51871	0.543775	0.271888	+	0.962329i	$0.412352\pi$
$104$	−5.26606	−0.516380
$105$	0	0
$106$	−11.2107	−1.08888
$107$	−13.4054	−1.29595	−0.647976	−	0.761660i	$-0.724385\pi$
$107$	−13.4054	−1.29595	−0.647976	+	0.761660i	$0.724385\pi$
$108$	0	0
$109$	−5.10329	−0.488806	−0.244403	−	0.969674i	$-0.578592\pi$
$109$	−5.10329	−0.488806	−0.244403	+	0.969674i	$0.578592\pi$
$110$	7.40934	0.706452
$111$	0	0
$112$	−3.32927	−0.314586
$113$	−0.905016	−0.0851368	−0.0425684	−	0.999094i	$-0.513554\pi$
$113$	−0.905016	−0.0851368	−0.0425684	+	0.999094i	$0.513554\pi$
$114$	0	0
$115$	−13.3220	−1.24228
$116$	25.1784	2.33775
$117$	0	0
$118$	−5.95773	−0.548454
$119$	−9.06871	−0.831327
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.16302	−0.195831
$123$	0	0
$124$	−15.7312	−1.41270
$125$	−5.93899	−0.531199
$126$	0	0
$127$	2.67303	0.237193	0.118597	−	0.992943i	$-0.462160\pi$
$127$	2.67303	0.237193	0.118597	+	0.992943i	$0.462160\pi$
$128$	11.0674	0.978229
$129$	0	0
$130$	26.5802	2.33124
$131$	−11.8629	−1.03647	−0.518235	−	0.855238i	$-0.673411\pi$
$131$	−11.8629	−1.03647	−0.518235	+	0.855238i	$0.673411\pi$
$132$	0	0
$133$	−2.52714	−0.219131
$134$	−15.7591	−1.36138
$135$	0	0
$136$	8.72538	0.748195
$137$	−20.2371	−1.72897	−0.864485	−	0.502658i	$-0.832356\pi$
$137$	−20.2371	−1.72897	−0.864485	+	0.502658i	$0.832356\pi$
$138$	0	0
$139$	−20.2174	−1.71482	−0.857410	−	0.514634i	$-0.827928\pi$
$139$	−20.2174	−1.71482	−0.857410	+	0.514634i	$0.827928\pi$
$140$	−13.9992	−1.18315
$141$	0	0
$142$	−28.1954	−2.36611
$143$	3.58739	0.299993
$144$	0	0
$145$	−32.1981	−2.67391
$146$	−5.29049	−0.437844
$147$	0	0
$148$	8.82932	0.725766
$149$	10.2507	0.839766	0.419883	−	0.907578i	$-0.362071\pi$
$149$	10.2507	0.839766	0.419883	+	0.907578i	$0.362071\pi$
$150$	0	0
$151$	−7.52745	−0.612575	−0.306288	−	0.951939i	$-0.599087\pi$
$151$	−7.52745	−0.612575	−0.306288	+	0.951939i	$0.599087\pi$
$152$	2.43147	0.197218
$153$	0	0
$154$	−3.30011	−0.265930
$155$	20.1170	1.61584
$156$	0	0
$157$	2.92328	0.233303	0.116652	−	0.993173i	$-0.462784\pi$
$157$	2.92328	0.233303	0.116652	+	0.993173i	$0.462784\pi$
$158$	32.5771	2.59170
$159$	0	0
$160$	−26.2248	−2.07326
$161$	5.93358	0.467632
$162$	0	0
$163$	−4.65218	−0.364387	−0.182193	−	0.983263i	$-0.558320\pi$
$163$	−4.65218	−0.364387	−0.182193	+	0.983263i	$0.558320\pi$
$164$	−30.1773	−2.35645
$165$	0	0
$166$	−22.4119	−1.73950
$167$	−9.20733	−0.712484	−0.356242	−	0.934394i	$-0.615942\pi$
$167$	−9.20733	−0.712484	−0.356242	+	0.934394i	$0.615942\pi$
$168$	0	0
$169$	−0.130600	−0.0100461
$170$	−44.0409	−3.37779
$171$	0	0
$172$	13.3034	1.01437
$173$	6.01897	0.457614	0.228807	−	0.973472i	$-0.426517\pi$
$173$	6.01897	0.457614	0.228807	+	0.973472i	$0.426517\pi$
$174$	0	0
$175$	10.2737	0.776619
$176$	−2.18213	−0.164484
$177$	0	0
$178$	0.902660	0.0676572
$179$	−11.4464	−0.855545	−0.427773	−	0.903886i	$-0.640702\pi$
$179$	−11.4464	−0.855545	−0.427773	+	0.903886i	$0.640702\pi$
$180$	0	0
$181$	19.2696	1.43230	0.716148	−	0.697948i	$-0.245904\pi$
$181$	19.2696	1.43230	0.716148	+	0.697948i	$0.245904\pi$
$182$	−11.8388	−0.877550
$183$	0	0
$184$	−5.70894	−0.420869
$185$	−11.2909	−0.830126
$186$	0	0
$187$	−5.94398	−0.434667
$188$	−1.88783	−0.137684
$189$	0	0
$190$	−12.2727	−0.890357
$191$	7.11913	0.515122	0.257561	−	0.966262i	$-0.417081\pi$
$191$	7.11913	0.515122	0.257561	+	0.966262i	$0.417081\pi$
$192$	0	0
$193$	25.2029	1.81415	0.907073	−	0.420973i	$-0.138311\pi$
$193$	25.2029	1.81415	0.907073	+	0.420973i	$0.138311\pi$
$194$	25.4584	1.82781
$195$	0	0
$196$	−12.5153	−0.893952
$197$	−20.3647	−1.45092	−0.725461	−	0.688263i	$-0.758374\pi$
$197$	−20.3647	−1.45092	−0.725461	+	0.688263i	$0.758374\pi$
$198$	0	0
$199$	13.8349	0.980729	0.490365	−	0.871517i	$-0.336864\pi$
$199$	13.8349	0.980729	0.490365	+	0.871517i	$0.336864\pi$
$200$	−9.88475	−0.698957
$201$	0	0
$202$	−30.2983	−2.13178
$203$	14.3410	1.00654
$204$	0	0
$205$	38.5908	2.69530
$206$	−11.9371	−0.831696
$207$	0	0
$208$	−7.82816	−0.542785
$209$	−1.65639	−0.114575
$210$	0	0
$211$	14.7335	1.01430	0.507149	−	0.861858i	$-0.330699\pi$
$211$	14.7335	1.01430	0.507149	+	0.861858i	$0.330699\pi$
$212$	13.8832	0.953503
$213$	0	0
$214$	28.9962	1.98214
$215$	−17.0124	−1.16023
$216$	0	0
$217$	−8.96009	−0.608251
$218$	11.0385	0.747622
$219$	0	0
$220$	−9.17561	−0.618620
$221$	−21.3234	−1.43437
$222$	0	0
$223$	25.0230	1.67566	0.837831	−	0.545929i	$-0.183823\pi$
$223$	25.0230	1.67566	0.837831	+	0.545929i	$0.183823\pi$
$224$	11.6805	0.780437
$225$	0	0
$226$	1.95757	0.130215
$227$	−14.9179	−0.990136	−0.495068	−	0.868854i	$-0.664857\pi$
$227$	−14.9179	−0.990136	−0.495068	+	0.868854i	$0.664857\pi$
$228$	0	0
$229$	13.0514	0.862459	0.431230	−	0.902242i	$-0.358080\pi$
$229$	13.0514	0.862459	0.431230	+	0.902242i	$0.358080\pi$
$230$	28.8156	1.90005
$231$	0	0
$232$	−13.7981	−0.905888
$233$	−25.8089	−1.69080	−0.845399	−	0.534135i	$-0.820637\pi$
$233$	−25.8089	−1.69080	−0.845399	+	0.534135i	$0.820637\pi$
$234$	0	0
$235$	2.41416	0.157483
$236$	7.37797	0.480265
$237$	0	0
$238$	19.6158	1.27150
$239$	26.6536	1.72408	0.862039	−	0.506841i	$-0.169187\pi$
$239$	26.6536	1.72408	0.862039	+	0.506841i	$0.169187\pi$
$240$	0	0
$241$	7.39439	0.476315	0.238157	−	0.971227i	$-0.423457\pi$
$241$	7.39439	0.476315	0.238157	+	0.971227i	$0.423457\pi$
$242$	−2.16302	−0.139044
$243$	0	0
$244$	2.67865	0.171483
$245$	16.0046	1.02250
$246$	0	0
$247$	−5.94212	−0.378088
$248$	8.62088	0.547426
$249$	0	0
$250$	12.8461	0.812462
$251$	−13.6943	−0.864374	−0.432187	−	0.901784i	$-0.642258\pi$
$251$	−13.6943	−0.864374	−0.432187	+	0.901784i	$0.642258\pi$
$252$	0	0
$253$	3.88910	0.244506
$254$	−5.78182	−0.362784
$255$	0	0
$256$	0.452025	0.0282515
$257$	−14.3062	−0.892395	−0.446197	−	0.894935i	$-0.647222\pi$
$257$	−14.3062	−0.892395	−0.446197	+	0.894935i	$0.647222\pi$
$258$	0	0
$259$	5.02897	0.312485
$260$	−32.9165	−2.04140
$261$	0	0
$262$	25.6598	1.58527
$263$	2.96892	0.183071	0.0915356	−	0.995802i	$-0.470822\pi$
$263$	2.96892	0.183071	0.0915356	+	0.995802i	$0.470822\pi$
$264$	0	0
$265$	−17.7539	−1.09061
$266$	5.46626	0.335158
$267$	0	0
$268$	19.5158	1.19212
$269$	−24.2725	−1.47992	−0.739960	−	0.672651i	$-0.765156\pi$
$269$	−24.2725	−1.47992	−0.739960	+	0.672651i	$0.765156\pi$
$270$	0	0
$271$	−14.2938	−0.868289	−0.434144	−	0.900843i	$-0.642949\pi$
$271$	−14.2938	−0.868289	−0.434144	+	0.900843i	$0.642949\pi$
$272$	12.9705	0.786455
$273$	0	0
$274$	43.7732	2.64444
$275$	6.73378	0.406062
$276$	0	0
$277$	−16.6354	−0.999525	−0.499762	−	0.866163i	$-0.666579\pi$
$277$	−16.6354	−0.999525	−0.499762	+	0.866163i	$0.666579\pi$
$278$	43.7307	2.62279
$279$	0	0
$280$	7.67174	0.458474
$281$	19.7970	1.18099	0.590496	−	0.807041i	$-0.298932\pi$
$281$	19.7970	1.18099	0.590496	+	0.807041i	$0.298932\pi$
$282$	0	0
$283$	−8.46139	−0.502977	−0.251489	−	0.967860i	$-0.580920\pi$
$283$	−8.46139	−0.502977	−0.251489	+	0.967860i	$0.580920\pi$
$284$	34.9168	2.07193
$285$	0	0
$286$	−7.75960	−0.458835
$287$	−17.1883	−1.01459
$288$	0	0
$289$	18.3309	1.07829
$290$	69.6451	4.08970
$291$	0	0
$292$	6.55167	0.383407
$293$	9.48622	0.554191	0.277096	−	0.960842i	$-0.410628\pi$
$293$	9.48622	0.554191	0.277096	+	0.960842i	$0.410628\pi$
$294$	0	0
$295$	−9.43495	−0.549324
$296$	−4.83858	−0.281237
$297$	0	0
$298$	−22.1724	−1.28441
$299$	13.9517	0.806849
$300$	0	0
$301$	7.57729	0.436748
$302$	16.2820	0.936925
$303$	0	0
$304$	3.61445	0.207303
$305$	−3.42546	−0.196141
$306$	0	0
$307$	14.4454	0.824443	0.412221	−	0.911084i	$-0.364753\pi$
$307$	14.4454	0.824443	0.412221	+	0.911084i	$0.364753\pi$
$308$	4.08681	0.232867
$309$	0	0
$310$	−43.5135	−2.47140
$311$	15.3587	0.870911	0.435455	−	0.900210i	$-0.356587\pi$
$311$	15.3587	0.870911	0.435455	+	0.900210i	$0.356587\pi$
$312$	0	0
$313$	−4.36876	−0.246937	−0.123469	−	0.992348i	$-0.539402\pi$
$313$	−4.36876	−0.246937	−0.123469	+	0.992348i	$0.539402\pi$
$314$	−6.32312	−0.356834
$315$	0	0
$316$	−40.3431	−2.26947
$317$	−5.20003	−0.292063	−0.146031	−	0.989280i	$-0.546650\pi$
$317$	−5.20003	−0.292063	−0.146031	+	0.989280i	$0.546650\pi$
$318$	0	0
$319$	9.39965	0.526279
$320$	41.7752	2.33531
$321$	0	0
$322$	−12.8345	−0.715236
$323$	9.84554	0.547820
$324$	0	0
$325$	24.1567	1.33997
$326$	10.0628	0.557324
$327$	0	0
$328$	16.5376	0.913134
$329$	−1.07527	−0.0592813
$330$	0	0
$331$	5.82973	0.320431	0.160216	−	0.987082i	$-0.448781\pi$
$331$	5.82973	0.320431	0.160216	+	0.987082i	$0.448781\pi$
$332$	27.7546	1.52323
$333$	0	0
$334$	19.9156	1.08973
$335$	−24.9569	−1.36354
$336$	0	0
$337$	−23.3465	−1.27176	−0.635882	−	0.771786i	$-0.719363\pi$
$337$	−23.3465	−1.27176	−0.635882	+	0.771786i	$0.719363\pi$
$338$	0.282490	0.0153654
$339$	0	0
$340$	54.5397	2.95783
$341$	−5.87279	−0.318029
$342$	0	0
$343$	−17.8083	−0.961558
$344$	−7.29042	−0.393073
$345$	0	0
$346$	−13.0192	−0.699914
$347$	21.7134	1.16563	0.582817	−	0.812603i	$-0.301950\pi$
$347$	21.7134	1.16563	0.582817	+	0.812603i	$0.301950\pi$
$348$	0	0
$349$	36.0189	1.92805	0.964023	−	0.265818i	$-0.0856420\pi$
$349$	36.0189	1.92805	0.964023	+	0.265818i	$0.0856420\pi$
$350$	−22.2222	−1.18783
$351$	0	0
$352$	7.65586	0.408059
$353$	3.13801	0.167019	0.0835097	−	0.996507i	$-0.473387\pi$
$353$	3.13801	0.167019	0.0835097	+	0.996507i	$0.473387\pi$
$354$	0	0
$355$	−44.6516	−2.36986
$356$	−1.11784	−0.0592455
$357$	0	0
$358$	24.7588	1.30854
$359$	−4.01251	−0.211772	−0.105886	−	0.994378i	$-0.533768\pi$
$359$	−4.01251	−0.211772	−0.105886	+	0.994378i	$0.533768\pi$
$360$	0	0
$361$	−16.2564	−0.855599
$362$	−41.6805	−2.19068
$363$	0	0
$364$	14.6610	0.768445
$365$	−8.37828	−0.438539
$366$	0	0
$367$	16.7099	0.872250	0.436125	−	0.899886i	$-0.356350\pi$
$367$	16.7099	0.872250	0.436125	+	0.899886i	$0.356350\pi$
$368$	−8.48652	−0.442390
$369$	0	0
$370$	24.4225	1.26967
$371$	7.90755	0.410539
$372$	0	0
$373$	26.5321	1.37378	0.686890	−	0.726761i	$-0.258975\pi$
$373$	26.5321	1.37378	0.686890	+	0.726761i	$0.258975\pi$
$374$	12.8569	0.664817
$375$	0	0
$376$	1.03456	0.0533532
$377$	33.7202	1.73668
$378$	0	0
$379$	5.35976	0.275312	0.137656	−	0.990480i	$-0.456043\pi$
$379$	5.35976	0.275312	0.137656	+	0.990480i	$0.456043\pi$
$380$	15.1984	0.779660
$381$	0	0
$382$	−15.3988	−0.787871
$383$	20.8095	1.06331	0.531657	−	0.846960i	$-0.321570\pi$
$383$	20.8095	1.06331	0.531657	+	0.846960i	$0.321570\pi$
$384$	0	0
$385$	−5.22621	−0.266352
$386$	−54.5144	−2.77471
$387$	0	0
$388$	−31.5273	−1.60056
$389$	5.32194	0.269833	0.134917	−	0.990857i	$-0.456923\pi$
$389$	5.32194	0.269833	0.134917	+	0.990857i	$0.456923\pi$
$390$	0	0
$391$	−23.1167	−1.16906
$392$	6.85856	0.346410
$393$	0	0
$394$	44.0492	2.21917
$395$	51.5907	2.59581
$396$	0	0
$397$	−29.6593	−1.48856	−0.744279	−	0.667869i	$-0.767207\pi$
$397$	−29.6593	−1.48856	−0.744279	+	0.667869i	$0.767207\pi$
$398$	−29.9251	−1.50001
$399$	0	0
$400$	−14.6940	−0.734699
$401$	25.5116	1.27399	0.636994	−	0.770869i	$-0.280177\pi$
$401$	25.5116	1.27399	0.636994	+	0.770869i	$0.280177\pi$
$402$	0	0
$403$	−21.0680	−1.04947
$404$	37.5210	1.86674
$405$	0	0
$406$	−31.0199	−1.53949
$407$	3.29618	0.163386
$408$	0	0
$409$	36.4764	1.80364	0.901822	−	0.432107i	$-0.142230\pi$
$409$	36.4764	1.80364	0.901822	+	0.432107i	$0.142230\pi$
$410$	−83.4725	−4.12241
$411$	0	0
$412$	14.7827	0.728292
$413$	4.20232	0.206782
$414$	0	0
$415$	−35.4926	−1.74226
$416$	27.4646	1.34656
$417$	0	0
$418$	3.58280	0.175240
$419$	−5.85835	−0.286199	−0.143099	−	0.989708i	$-0.545707\pi$
$419$	−5.85835	−0.286199	−0.143099	+	0.989708i	$0.545707\pi$
$420$	0	0
$421$	1.05938	0.0516310	0.0258155	−	0.999667i	$-0.491782\pi$
$421$	1.05938	0.0516310	0.0258155	+	0.999667i	$0.491782\pi$
$422$	−31.8689	−1.55135
$423$	0	0
$424$	−7.60818	−0.369486
$425$	−40.0254	−1.94152
$426$	0	0
$427$	1.52570	0.0738336
$428$	−35.9085	−1.73570
$429$	0	0
$430$	36.7981	1.77456
$431$	7.71977	0.371848	0.185924	−	0.982564i	$-0.440472\pi$
$431$	7.71977	0.371848	0.185924	+	0.982564i	$0.440472\pi$
$432$	0	0
$433$	2.29563	0.110321	0.0551605	−	0.998477i	$-0.482433\pi$
$433$	2.29563	0.110321	0.0551605	+	0.998477i	$0.482433\pi$
$434$	19.3809	0.930311
$435$	0	0
$436$	−13.6699	−0.654671
$437$	−6.44185	−0.308156
$438$	0	0
$439$	−11.8920	−0.567573	−0.283787	−	0.958887i	$-0.591591\pi$
$439$	−11.8920	−0.567573	−0.283787	+	0.958887i	$0.591591\pi$
$440$	5.02835	0.239717
$441$	0	0
$442$	46.1229	2.19384
$443$	6.93619	0.329548	0.164774	−	0.986331i	$-0.447310\pi$
$443$	6.93619	0.329548	0.164774	+	0.986331i	$0.447310\pi$
$444$	0	0
$445$	1.42950	0.0677646
$446$	−54.1252	−2.56290
$447$	0	0
$448$	−18.6066	−0.879081
$449$	−24.3086	−1.14719	−0.573597	−	0.819137i	$-0.694453\pi$
$449$	−24.3086	−1.14719	−0.573597	+	0.819137i	$0.694453\pi$
$450$	0	0
$451$	−11.2659	−0.530489
$452$	−2.42422	−0.114026
$453$	0	0
$454$	32.2677	1.51440
$455$	−18.7485	−0.878942
$456$	0	0
$457$	19.1536	0.895968	0.447984	−	0.894041i	$-0.352142\pi$
$457$	19.1536	0.895968	0.447984	+	0.894041i	$0.352142\pi$
$458$	−28.2304	−1.31912
$459$	0	0
$460$	−35.6849	−1.66382
$461$	21.0634	0.981019	0.490509	−	0.871436i	$-0.336811\pi$
$461$	21.0634	0.981019	0.490509	+	0.871436i	$0.336811\pi$
$462$	0	0
$463$	35.7967	1.66361	0.831807	−	0.555064i	$-0.187306\pi$
$463$	35.7967	1.66361	0.831807	+	0.555064i	$0.187306\pi$
$464$	−20.5113	−0.952211
$465$	0	0
$466$	55.8252	2.58605
$467$	9.34479	0.432425	0.216213	−	0.976346i	$-0.430630\pi$
$467$	9.34479	0.432425	0.216213	+	0.976346i	$0.430630\pi$
$468$	0	0
$469$	11.1158	0.513278
$470$	−5.22188	−0.240867
$471$	0	0
$472$	−4.04322	−0.186104
$473$	4.96645	0.228357
$474$	0	0
$475$	−11.1537	−0.511769
$476$	−24.2919	−1.11342
$477$	0	0
$478$	−57.6523	−2.63695
$479$	−22.2054	−1.01459	−0.507296	−	0.861772i	$-0.669355\pi$
$479$	−22.2054	−1.01459	−0.507296	+	0.861772i	$0.669355\pi$
$480$	0	0
$481$	11.8247	0.539160
$482$	−15.9942	−0.728517
$483$	0	0
$484$	2.67865	0.121757
$485$	40.3172	1.83071
$486$	0	0
$487$	4.44828	0.201571	0.100785	−	0.994908i	$-0.467864\pi$
$487$	4.44828	0.201571	0.100785	+	0.994908i	$0.467864\pi$
$488$	−1.46794	−0.0664503
$489$	0	0
$490$	−34.6183	−1.56389
$491$	16.3927	0.739791	0.369896	−	0.929073i	$-0.379393\pi$
$491$	16.3927	0.739791	0.369896	+	0.929073i	$0.379393\pi$
$492$	0	0
$493$	−55.8713	−2.51632
$494$	12.8529	0.578280
$495$	0	0
$496$	12.8152	0.575419
$497$	19.8878	0.892089
$498$	0	0
$499$	37.4385	1.67598	0.837989	−	0.545687i	$-0.183731\pi$
$499$	37.4385	1.67598	0.837989	+	0.545687i	$0.183731\pi$
$500$	−15.9085	−0.711449
$501$	0	0
$502$	29.6209	1.32205
$503$	16.2915	0.726403	0.363202	−	0.931711i	$-0.381684\pi$
$503$	16.2915	0.726403	0.363202	+	0.931711i	$0.381684\pi$
$504$	0	0
$505$	−47.9819	−2.13516
$506$	−8.41219	−0.373968
$507$	0	0
$508$	7.16012	0.317679
$509$	21.2376	0.941338	0.470669	−	0.882310i	$-0.344013\pi$
$509$	21.2376	0.941338	0.470669	+	0.882310i	$0.344013\pi$
$510$	0	0
$511$	3.73167	0.165080
$512$	−23.1125	−1.02144
$513$	0	0
$514$	30.9445	1.36490
$515$	−18.9041	−0.833016
$516$	0	0
$517$	−0.704770	−0.0309958
$518$	−10.8778	−0.477941
$519$	0	0
$520$	18.0387	0.791049
$521$	−25.0948	−1.09942	−0.549711	−	0.835355i	$-0.685262\pi$
$521$	−25.0948	−1.09942	−0.549711	+	0.835355i	$0.685262\pi$
$522$	0	0
$523$	7.35803	0.321744	0.160872	−	0.986975i	$-0.448569\pi$
$523$	7.35803	0.321744	0.160872	+	0.986975i	$0.448569\pi$
$524$	−31.7767	−1.38817
$525$	0	0
$526$	−6.42182	−0.280005
$527$	34.9077	1.52061
$528$	0	0
$529$	−7.87492	−0.342388
$530$	38.4019	1.66807
$531$	0	0
$532$	−6.76934	−0.293488
$533$	−40.4151	−1.75057
$534$	0	0
$535$	45.9198	1.98529
$536$	−10.6949	−0.461950
$537$	0	0
$538$	52.5019	2.26352
$539$	−4.67225	−0.201248
$540$	0	0
$541$	−21.6515	−0.930869	−0.465434	−	0.885082i	$-0.654102\pi$
$541$	−21.6515	−0.930869	−0.465434	+	0.885082i	$0.654102\pi$
$542$	30.9178	1.32803
$543$	0	0
$544$	−45.5063	−1.95107
$545$	17.4811	0.748809
$546$	0	0
$547$	21.6436	0.925413	0.462706	−	0.886512i	$-0.346878\pi$
$547$	21.6436	0.925413	0.462706	+	0.886512i	$0.346878\pi$
$548$	−54.2081	−2.31565
$549$	0	0
$550$	−14.5653	−0.621066
$551$	−15.5695	−0.663281
$552$	0	0
$553$	−22.9785	−0.977143
$554$	35.9827	1.52876
$555$	0	0
$556$	−54.1554	−2.29670
$557$	5.39479	0.228585	0.114292	−	0.993447i	$-0.463540\pi$
$557$	5.39479	0.228585	0.114292	+	0.993447i	$0.463540\pi$
$558$	0	0
$559$	17.8166	0.753562
$560$	11.4043	0.481919
$561$	0	0
$562$	−42.8213	−1.80631
$563$	42.3802	1.78611	0.893056	−	0.449945i	$-0.148556\pi$
$563$	42.3802	1.78611	0.893056	+	0.449945i	$0.148556\pi$
$564$	0	0
$565$	3.10010	0.130422
$566$	18.3021	0.769297
$567$	0	0
$568$	−19.1349	−0.802881
$569$	14.2313	0.596609	0.298304	−	0.954471i	$-0.403579\pi$
$569$	14.2313	0.596609	0.298304	+	0.954471i	$0.403579\pi$
$570$	0	0
$571$	13.9607	0.584237	0.292118	−	0.956382i	$-0.405640\pi$
$571$	13.9607	0.584237	0.292118	+	0.956382i	$0.405640\pi$
$572$	9.60938	0.401788
$573$	0	0
$574$	37.1786	1.55180
$575$	26.1883	1.09213
$576$	0	0
$577$	4.17275	0.173714	0.0868568	−	0.996221i	$-0.472318\pi$
$577$	4.17275	0.173714	0.0868568	+	0.996221i	$0.472318\pi$
$578$	−39.6501	−1.64923
$579$	0	0
$580$	−86.2475	−3.58123
$581$	15.8084	0.655841
$582$	0	0
$583$	5.18291	0.214654
$584$	−3.59040	−0.148572
$585$	0	0
$586$	−20.5189	−0.847627
$587$	27.1859	1.12208	0.561040	−	0.827789i	$-0.310401\pi$
$587$	27.1859	1.12208	0.561040	+	0.827789i	$0.310401\pi$
$588$	0	0
$589$	9.72762	0.400819
$590$	20.4080	0.840183
$591$	0	0
$592$	−7.19270	−0.295618
$593$	33.9524	1.39426	0.697130	−	0.716945i	$-0.254460\pi$
$593$	33.9524	1.39426	0.697130	+	0.716945i	$0.254460\pi$
$594$	0	0
$595$	31.0645	1.27352
$596$	27.4579	1.12472
$597$	0	0
$598$	−30.1779	−1.23406
$599$	−16.5015	−0.674234	−0.337117	−	0.941463i	$-0.609452\pi$
$599$	−16.5015	−0.674234	−0.337117	+	0.941463i	$0.609452\pi$
$600$	0	0
$601$	11.5317	0.470389	0.235195	−	0.971948i	$-0.424427\pi$
$601$	11.5317	0.470389	0.235195	+	0.971948i	$0.424427\pi$
$602$	−16.3898	−0.667999
$603$	0	0
$604$	−20.1634	−0.820437
$605$	−3.42546	−0.139265
$606$	0	0
$607$	−22.4643	−0.911799	−0.455899	−	0.890031i	$-0.650682\pi$
$607$	−22.4643	−0.911799	−0.455899	+	0.890031i	$0.650682\pi$
$608$	−12.6811	−0.514285
$609$	0	0
$610$	7.40934	0.299995
$611$	−2.52829	−0.102284
$612$	0	0
$613$	27.6946	1.11858	0.559288	−	0.828973i	$-0.311075\pi$
$613$	27.6946	1.11858	0.559288	+	0.828973i	$0.311075\pi$
$614$	−31.2457	−1.26097
$615$	0	0
$616$	−2.23962	−0.0902370
$617$	−1.45117	−0.0584219	−0.0292109	−	0.999573i	$-0.509299\pi$
$617$	−1.45117	−0.0584219	−0.0292109	+	0.999573i	$0.509299\pi$
$618$	0	0
$619$	−5.38440	−0.216417	−0.108209	−	0.994128i	$-0.534511\pi$
$619$	−5.38440	−0.216417	−0.108209	+	0.994128i	$0.534511\pi$
$620$	53.8864	2.16413
$621$	0	0
$622$	−33.2211	−1.33205
$623$	−0.636696	−0.0255087
$624$	0	0
$625$	−13.3251	−0.533005
$626$	9.44972	0.377687
$627$	0	0
$628$	7.83046	0.312469
$629$	−19.5924	−0.781202
$630$	0	0
$631$	36.7780	1.46411	0.732054	−	0.681247i	$-0.238562\pi$
$631$	36.7780	1.46411	0.732054	+	0.681247i	$0.238562\pi$
$632$	22.1085	0.879430
$633$	0	0
$634$	11.2478	0.446706
$635$	−9.15637	−0.363359
$636$	0	0
$637$	−16.7612	−0.664103
$638$	−20.3316	−0.804936
$639$	0	0
$640$	−37.9109	−1.49856
$641$	−8.54776	−0.337616	−0.168808	−	0.985649i	$-0.553992\pi$
$641$	−8.54776	−0.337616	−0.168808	+	0.985649i	$0.553992\pi$
$642$	0	0
$643$	−14.3390	−0.565475	−0.282737	−	0.959197i	$-0.591242\pi$
$643$	−14.3390	−0.565475	−0.282737	+	0.959197i	$0.591242\pi$
$644$	15.8940	0.626311
$645$	0	0
$646$	−21.2961	−0.837883
$647$	−23.0276	−0.905308	−0.452654	−	0.891686i	$-0.649523\pi$
$647$	−23.0276	−0.905308	−0.452654	+	0.891686i	$0.649523\pi$
$648$	0	0
$649$	2.75436	0.108118
$650$	−52.2514	−2.04947
$651$	0	0
$652$	−12.4616	−0.488033
$653$	21.8911	0.856666	0.428333	−	0.903621i	$-0.359101\pi$
$653$	21.8911	0.856666	0.428333	+	0.903621i	$0.359101\pi$
$654$	0	0
$655$	40.6361	1.58778
$656$	24.5836	0.959827
$657$	0	0
$658$	2.32582	0.0906699
$659$	40.5421	1.57930	0.789648	−	0.613560i	$-0.210263\pi$
$659$	40.5421	1.57930	0.789648	+	0.613560i	$0.210263\pi$
$660$	0	0
$661$	−32.1044	−1.24872	−0.624358	−	0.781138i	$-0.714639\pi$
$661$	−32.1044	−1.24872	−0.624358	+	0.781138i	$0.714639\pi$
$662$	−12.6098	−0.490095
$663$	0	0
$664$	−15.2099	−0.590258
$665$	8.65663	0.335690
$666$	0	0
$667$	36.5561	1.41546
$668$	−24.6632	−0.954249
$669$	0	0
$670$	53.9821	2.08551
$671$	1.00000	0.0386046
$672$	0	0
$673$	23.8166	0.918061	0.459030	−	0.888421i	$-0.348197\pi$
$673$	23.8166	0.918061	0.459030	+	0.888421i	$0.348197\pi$
$674$	50.4989	1.94514
$675$	0	0
$676$	−0.349831	−0.0134550
$677$	6.82508	0.262309	0.131155	−	0.991362i	$-0.458132\pi$
$677$	6.82508	0.262309	0.131155	+	0.991362i	$0.458132\pi$
$678$	0	0
$679$	−17.9572	−0.689135
$680$	−29.8884	−1.14617
$681$	0	0
$682$	12.7030	0.486421
$683$	−22.7708	−0.871301	−0.435651	−	0.900116i	$-0.643482\pi$
$683$	−22.7708	−0.871301	−0.435651	+	0.900116i	$0.643482\pi$
$684$	0	0
$685$	69.3213	2.64863
$686$	38.5197	1.47069
$687$	0	0
$688$	−10.8374	−0.413173
$689$	18.5931	0.708342
$690$	0	0
$691$	25.6291	0.974978	0.487489	−	0.873129i	$-0.337913\pi$
$691$	25.6291	0.974978	0.487489	+	0.873129i	$0.337913\pi$
$692$	16.1227	0.612894
$693$	0	0
$694$	−46.9664	−1.78282
$695$	69.2540	2.62695
$696$	0	0
$697$	66.9641	2.53644
$698$	−77.9095	−2.94892
$699$	0	0
$700$	27.5197	1.04015
$701$	39.5842	1.49507	0.747537	−	0.664220i	$-0.231236\pi$
$701$	39.5842	1.49507	0.747537	+	0.664220i	$0.231236\pi$
$702$	0	0
$703$	−5.45975	−0.205919
$704$	−12.1955	−0.459636
$705$	0	0
$706$	−6.78757	−0.255454
$707$	21.3711	0.803742
$708$	0	0
$709$	−47.7140	−1.79194	−0.895969	−	0.444117i	$-0.853518\pi$
$709$	−47.7140	−1.79194	−0.895969	+	0.444117i	$0.853518\pi$
$710$	96.5823	3.62467
$711$	0	0
$712$	0.612591	0.0229578
$713$	−22.8399	−0.855359
$714$	0	0
$715$	−12.2885	−0.459563
$716$	−30.6610	−1.14585
$717$	0	0
$718$	8.67914	0.323903
$719$	−6.77246	−0.252570	−0.126285	−	0.991994i	$-0.540305\pi$
$719$	−6.77246	−0.252570	−0.126285	+	0.991994i	$0.540305\pi$
$720$	0	0
$721$	8.41988	0.313573
$722$	35.1629	1.30863
$723$	0	0
$724$	51.6165	1.91831
$725$	63.2951	2.35072
$726$	0	0
$727$	−10.5969	−0.393017	−0.196509	−	0.980502i	$-0.562960\pi$
$727$	−10.5969	−0.393017	−0.196509	+	0.980502i	$0.562960\pi$
$728$	−8.03441	−0.297775
$729$	0	0
$730$	18.1224	0.670739
$731$	−29.5205	−1.09185
$732$	0	0
$733$	−13.7993	−0.509690	−0.254845	−	0.966982i	$-0.582025\pi$
$733$	−13.7993	−0.509690	−0.254845	+	0.966982i	$0.582025\pi$
$734$	−36.1439	−1.33409
$735$	0	0
$736$	29.7744	1.09750
$737$	7.28569	0.268372
$738$	0	0
$739$	19.6135	0.721493	0.360746	−	0.932664i	$-0.382522\pi$
$739$	19.6135	0.721493	0.360746	+	0.932664i	$0.382522\pi$
$740$	−30.2445	−1.11181
$741$	0	0
$742$	−17.1042	−0.627914
$743$	42.1891	1.54777	0.773884	−	0.633327i	$-0.218311\pi$
$743$	42.1891	1.54777	0.773884	+	0.633327i	$0.218311\pi$
$744$	0	0
$745$	−35.1132	−1.28645
$746$	−57.3895	−2.10118
$747$	0	0
$748$	−15.9219	−0.582161
$749$	−20.4526	−0.747323
$750$	0	0
$751$	37.1798	1.35671	0.678355	−	0.734734i	$-0.262693\pi$
$751$	37.1798	1.35671	0.678355	+	0.734734i	$0.262693\pi$
$752$	1.53790	0.0560815
$753$	0	0
$754$	−72.9375	−2.65623
$755$	25.7850	0.938411
$756$	0	0
$757$	−1.57018	−0.0570691	−0.0285345	−	0.999593i	$-0.509084\pi$
$757$	−1.57018	−0.0570691	−0.0285345	+	0.999593i	$0.509084\pi$
$758$	−11.5933	−0.421086
$759$	0	0
$760$	−8.32890	−0.302121
$761$	−20.3178	−0.736519	−0.368259	−	0.929723i	$-0.620046\pi$
$761$	−20.3178	−0.736519	−0.368259	+	0.929723i	$0.620046\pi$
$762$	0	0
$763$	−7.78607	−0.281875
$764$	19.0697	0.689916
$765$	0	0
$766$	−45.0112	−1.62632
$767$	9.88098	0.356781
$768$	0	0
$769$	15.8861	0.572868	0.286434	−	0.958100i	$-0.407530\pi$
$769$	15.8861	0.572868	0.286434	+	0.958100i	$0.407530\pi$
$770$	11.3044	0.407382
$771$	0	0
$772$	67.5098	2.42973
$773$	−53.2900	−1.91671	−0.958354	−	0.285582i	$-0.907813\pi$
$773$	−53.2900	−1.91671	−0.958354	+	0.285582i	$0.907813\pi$
$774$	0	0
$775$	−39.5461	−1.42054
$776$	17.2774	0.620222
$777$	0	0
$778$	−11.5115	−0.412706
$779$	18.6606	0.668586
$780$	0	0
$781$	13.0352	0.466437
$782$	50.0019	1.78806
$783$	0	0
$784$	10.1955	0.364124
$785$	−10.0136	−0.357400
$786$	0	0
$787$	30.4125	1.08409	0.542044	−	0.840350i	$-0.317651\pi$
$787$	30.4125	1.08409	0.542044	+	0.840350i	$0.317651\pi$
$788$	−54.5499	−1.94326
$789$	0	0
$790$	−111.592	−3.97026
$791$	−1.38078	−0.0490949
$792$	0	0
$793$	3.58739	0.127392
$794$	64.1537	2.27673
$795$	0	0
$796$	37.0588	1.31352
$797$	17.0007	0.602194	0.301097	−	0.953594i	$-0.402647\pi$
$797$	17.0007	0.602194	0.301097	+	0.953594i	$0.402647\pi$
$798$	0	0
$799$	4.18914	0.148201
$800$	51.5529	1.82267
$801$	0	0
$802$	−55.1821	−1.94855
$803$	2.44588	0.0863133
$804$	0	0
$805$	−20.3252	−0.716371
$806$	45.5705	1.60515
$807$	0	0
$808$	−20.5620	−0.723368
$809$	35.1744	1.23667	0.618333	−	0.785916i	$-0.287808\pi$
$809$	35.1744	1.23667	0.618333	+	0.785916i	$0.287808\pi$
$810$	0	0
$811$	1.02516	0.0359982	0.0179991	−	0.999838i	$-0.494270\pi$
$811$	1.02516	0.0359982	0.0179991	+	0.999838i	$0.494270\pi$
$812$	38.4145	1.34809
$813$	0	0
$814$	−7.12970	−0.249896
$815$	15.9359	0.558209
$816$	0	0
$817$	−8.22636	−0.287804
$818$	−78.8993	−2.75865
$819$	0	0
$820$	103.371	3.60988
$821$	8.69425	0.303431	0.151716	−	0.988424i	$-0.451520\pi$
$821$	8.69425	0.303431	0.151716	+	0.988424i	$0.451520\pi$
$822$	0	0
$823$	−9.37576	−0.326819	−0.163409	−	0.986558i	$-0.552249\pi$
$823$	−9.37576	−0.326819	−0.163409	+	0.986558i	$0.552249\pi$
$824$	−8.10111	−0.282216
$825$	0	0
$826$	−9.08969	−0.316271
$827$	−0.250732	−0.00871880	−0.00435940	−	0.999990i	$-0.501388\pi$
$827$	−0.250732	−0.00871880	−0.00435940	+	0.999990i	$0.501388\pi$
$828$	0	0
$829$	21.2866	0.739316	0.369658	−	0.929168i	$-0.379475\pi$
$829$	21.2866	0.739316	0.369658	+	0.929168i	$0.379475\pi$
$830$	76.7712	2.66477
$831$	0	0
$832$	−43.7501	−1.51676
$833$	27.7718	0.962235
$834$	0	0
$835$	31.5393	1.09146
$836$	−4.43688	−0.153453
$837$	0	0
$838$	12.6717	0.437737
$839$	21.3671	0.737674	0.368837	−	0.929494i	$-0.379756\pi$
$839$	21.3671	0.737674	0.368837	+	0.929494i	$0.379756\pi$
$840$	0	0
$841$	59.3533	2.04667
$842$	−2.29146	−0.0789688
$843$	0	0
$844$	39.4660	1.35847
$845$	0.447364	0.0153898
$846$	0	0
$847$	1.52570	0.0524235
$848$	−11.3098	−0.388380
$849$	0	0
$850$	86.5758	2.96953
$851$	12.8192	0.439436
$852$	0	0
$853$	−8.32954	−0.285198	−0.142599	−	0.989781i	$-0.545546\pi$
$853$	−8.32954	−0.285198	−0.142599	+	0.989781i	$0.545546\pi$
$854$	−3.30011	−0.112927
$855$	0	0
$856$	19.6783	0.672591
$857$	−16.3095	−0.557123	−0.278562	−	0.960418i	$-0.589858\pi$
$857$	−16.3095	−0.557123	−0.278562	+	0.960418i	$0.589858\pi$
$858$	0	0
$859$	−22.0010	−0.750665	−0.375332	−	0.926890i	$-0.622471\pi$
$859$	−22.0010	−0.750665	−0.375332	+	0.926890i	$0.622471\pi$
$860$	−45.5702	−1.55393
$861$	0	0
$862$	−16.6980	−0.568737
$863$	11.0086	0.374736	0.187368	−	0.982290i	$-0.440004\pi$
$863$	11.0086	0.374736	0.187368	+	0.982290i	$0.440004\pi$
$864$	0	0
$865$	−20.6178	−0.701025
$866$	−4.96550	−0.168734
$867$	0	0
$868$	−24.0010	−0.814646
$869$	−15.0610	−0.510908
$870$	0	0
$871$	26.1367	0.885607
$872$	7.49130	0.253687
$873$	0	0
$874$	13.9339	0.471320
$875$	−9.06109	−0.306321
$876$	0	0
$877$	−44.9955	−1.51939	−0.759694	−	0.650280i	$-0.774651\pi$
$877$	−44.9955	−1.51939	−0.759694	+	0.650280i	$0.774651\pi$
$878$	25.7226	0.868095
$879$	0	0
$880$	7.47480	0.251975
$881$	−30.3369	−1.02208	−0.511038	−	0.859558i	$-0.670739\pi$
$881$	−30.3369	−1.02208	−0.511038	+	0.859558i	$0.670739\pi$
$882$	0	0
$883$	49.0282	1.64993	0.824964	−	0.565185i	$-0.191195\pi$
$883$	49.0282	1.64993	0.824964	+	0.565185i	$0.191195\pi$
$884$	−57.1180	−1.92108
$885$	0	0
$886$	−15.0031	−0.504039
$887$	29.5968	0.993765	0.496882	−	0.867818i	$-0.334478\pi$
$887$	29.5968	0.993765	0.496882	+	0.867818i	$0.334478\pi$
$888$	0	0
$889$	4.07824	0.136780
$890$	−3.09203	−0.103645
$891$	0	0
$892$	67.0278	2.24426
$893$	1.16737	0.0390646
$894$	0	0
$895$	39.2093	1.31062
$896$	16.8855	0.564105
$897$	0	0
$898$	52.5800	1.75462
$899$	−55.2021	−1.84109
$900$	0	0
$901$	−30.8071	−1.02633
$902$	24.3683	0.811375
$903$	0	0
$904$	1.32851	0.0441854
$905$	−66.0072	−2.19415
$906$	0	0
$907$	−50.4919	−1.67656	−0.838278	−	0.545243i	$-0.816438\pi$
$907$	−50.4919	−1.67656	−0.838278	+	0.545243i	$0.816438\pi$
$908$	−39.9599	−1.32611
$909$	0	0
$910$	40.5533	1.34433
$911$	−4.74832	−0.157319	−0.0786594	−	0.996902i	$-0.525064\pi$
$911$	−4.74832	−0.157319	−0.0786594	+	0.996902i	$0.525064\pi$
$912$	0	0
$913$	10.3614	0.342913
$914$	−41.4296	−1.37037
$915$	0	0
$916$	34.9601	1.15511
$917$	−18.0993	−0.597690
$918$	0	0
$919$	−1.12645	−0.0371581	−0.0185791	−	0.999827i	$-0.505914\pi$
$919$	−1.12645	−0.0371581	−0.0185791	+	0.999827i	$0.505914\pi$
$920$	19.5558	0.644734
$921$	0	0
$922$	−45.5605	−1.50045
$923$	46.7625	1.53921
$924$	0	0
$925$	22.1958	0.729792
$926$	−77.4290	−2.54447
$927$	0	0
$928$	71.9624	2.36228
$929$	33.0322	1.08375	0.541876	−	0.840458i	$-0.317714\pi$
$929$	33.0322	1.08375	0.541876	+	0.840458i	$0.317714\pi$
$930$	0	0
$931$	7.73906	0.253637
$932$	−69.1331	−2.26453
$933$	0	0
$934$	−20.2130	−0.661388
$935$	20.3609	0.665872
$936$	0	0
$937$	−3.78011	−0.123491	−0.0617454	−	0.998092i	$-0.519667\pi$
$937$	−3.78011	−0.123491	−0.0617454	+	0.998092i	$0.519667\pi$
$938$	−24.0436	−0.785051
$939$	0	0
$940$	6.46670	0.210920
$941$	37.2259	1.21353	0.606764	−	0.794882i	$-0.292467\pi$
$941$	37.2259	1.21353	0.606764	+	0.794882i	$0.292467\pi$
$942$	0	0
$943$	−43.8140	−1.42678
$944$	−6.01037	−0.195621
$945$	0	0
$946$	−10.7425	−0.349269
$947$	−4.77277	−0.155094	−0.0775472	−	0.996989i	$-0.524709\pi$
$947$	−4.77277	−0.155094	−0.0775472	+	0.996989i	$0.524709\pi$
$948$	0	0
$949$	8.77435	0.284827
$950$	24.1258	0.782743
$951$	0	0
$952$	13.3123	0.431453
$953$	−42.2165	−1.36753	−0.683764	−	0.729703i	$-0.739658\pi$
$953$	−42.2165	−1.36753	−0.683764	+	0.729703i	$0.739658\pi$
$954$	0	0
$955$	−24.3863	−0.789122
$956$	71.3957	2.30910
$957$	0	0
$958$	48.0308	1.55180
$959$	−30.8756	−0.997026
$960$	0	0
$961$	3.48966	0.112570
$962$	−25.5771	−0.824638
$963$	0	0
$964$	19.8070	0.637941
$965$	−86.3316	−2.77911
$966$	0	0
$967$	13.6122	0.437740	0.218870	−	0.975754i	$-0.429763\pi$
$967$	13.6122	0.437740	0.218870	+	0.975754i	$0.429763\pi$
$968$	−1.46794	−0.0471812
$969$	0	0
$970$	−87.2068	−2.80004
$971$	57.5696	1.84750	0.923748	−	0.383001i	$-0.125109\pi$
$971$	57.5696	1.84750	0.923748	+	0.383001i	$0.125109\pi$
$972$	0	0
$973$	−30.8457	−0.988866
$974$	−9.62172	−0.308300
$975$	0	0
$976$	−2.18213	−0.0698483
$977$	0.417882	0.0133692	0.00668462	−	0.999978i	$-0.497872\pi$
$977$	0.417882	0.0133692	0.00668462	+	0.999978i	$0.497872\pi$
$978$	0	0
$979$	−0.417315	−0.0133374
$980$	42.8708	1.36946
$981$	0	0
$982$	−35.4577	−1.13150
$983$	−40.5086	−1.29202	−0.646012	−	0.763328i	$-0.723564\pi$
$983$	−40.5086	−1.29202	−0.646012	+	0.763328i	$0.723564\pi$
$984$	0	0
$985$	69.7584	2.22269
$986$	120.851	3.84867
$987$	0	0
$988$	−15.9169	−0.506383
$989$	19.3150	0.614181
$990$	0	0
$991$	38.3965	1.21971	0.609853	−	0.792514i	$-0.291228\pi$
$991$	38.3965	1.21971	0.609853	+	0.792514i	$0.291228\pi$
$992$	−44.9613	−1.42752
$993$	0	0
$994$	−43.0176	−1.36444
$995$	−47.3909	−1.50239
$996$	0	0
$997$	−8.37560	−0.265258	−0.132629	−	0.991166i	$-0.542342\pi$
$997$	−8.37560	−0.265258	−0.132629	+	0.991166i	$0.542342\pi$
$998$	−80.9802	−2.56338
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.l.1.4		21
3.2	odd	2		671.2.a.d.1.18	✓	21
33.32	even	2		7381.2.a.j.1.4		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.d.1.18	✓	21	3.2	odd	2
6039.2.a.l.1.4		21	1.1	even	1	trivial
7381.2.a.j.1.4		21	33.32	even	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.16302 q^{2} +2.67865 q^{4} -3.42546 q^{5} +1.52570 q^{7} -1.46794 q^{8} +O(q^{10})\)	\(q-2.16302 q^{2} +2.67865 q^{4} -3.42546 q^{5} +1.52570 q^{7} -1.46794 q^{8} +7.40934 q^{10} +1.00000 q^{11} +3.58739 q^{13} -3.30011 q^{14} -2.18213 q^{16} -5.94398 q^{17} -1.65639 q^{19} -9.17561 q^{20} -2.16302 q^{22} +3.88910 q^{23} +6.73378 q^{25} -7.75960 q^{26} +4.08681 q^{28} +9.39965 q^{29} -5.87279 q^{31} +7.65586 q^{32} +12.8569 q^{34} -5.22621 q^{35} +3.29618 q^{37} +3.58280 q^{38} +5.02835 q^{40} -11.2659 q^{41} +4.96645 q^{43} +2.67865 q^{44} -8.41219 q^{46} -0.704770 q^{47} -4.67225 q^{49} -14.5653 q^{50} +9.60938 q^{52} +5.18291 q^{53} -3.42546 q^{55} -2.23962 q^{56} -20.3316 q^{58} +2.75436 q^{59} +1.00000 q^{61} +12.7030 q^{62} -12.1955 q^{64} -12.2885 q^{65} +7.28569 q^{67} -15.9219 q^{68} +11.3044 q^{70} +13.0352 q^{71} +2.44588 q^{73} -7.12970 q^{74} -4.43688 q^{76} +1.52570 q^{77} -15.0610 q^{79} +7.47480 q^{80} +24.3683 q^{82} +10.3614 q^{83} +20.3609 q^{85} -10.7425 q^{86} -1.46794 q^{88} -0.417315 q^{89} +5.47327 q^{91} +10.4175 q^{92} +1.52443 q^{94} +5.67389 q^{95} -11.7699 q^{97} +10.1062 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) \) 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8	\( 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8} + q^{10} + 21 q^{11} + 20 q^{13} - 17 q^{14} + 50 q^{16} - q^{17} + 15 q^{19} + 2 q^{20} - 11 q^{23} + 48 q^{25} + 5 q^{26} - 16 q^{28} + 9 q^{29} + 22 q^{31} - 3 q^{32} + 33 q^{34} + 39 q^{35} + 21 q^{37} - 11 q^{38} - 16 q^{40} - 7 q^{41} + 16 q^{43} + 32 q^{44} - 3 q^{46} - 5 q^{47} + 80 q^{49} + 33 q^{50} + 60 q^{52} - 9 q^{53} - 7 q^{55} - 44 q^{56} - 27 q^{58} - 13 q^{59} + 21 q^{61} + 23 q^{62} + 66 q^{64} - 25 q^{65} + 38 q^{67} + 74 q^{68} - 33 q^{70} - 12 q^{71} + 20 q^{73} + 12 q^{74} + 59 q^{76} + 5 q^{77} + q^{79} + 38 q^{80} + 7 q^{82} + 19 q^{83} + 38 q^{85} + 3 q^{86} + 6 q^{88} - 37 q^{89} + 24 q^{91} - 31 q^{92} - 64 q^{94} + 43 q^{95} + 68 q^{97} + 2 q^{98}+O(q^{100}) \) 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8 + q^10 + 21 * q^11 + 20 * q^13 - 17 * q^14 + 50 * q^16 - q^17 + 15 * q^19 + 2 * q^20 - 11 * q^23 + 48 * q^25 + 5 * q^26 - 16 * q^28 + 9 * q^29 + 22 * q^31 - 3 * q^32 + 33 * q^34 + 39 * q^35 + 21 * q^37 - 11 * q^38 - 16 * q^40 - 7 * q^41 + 16 * q^43 + 32 * q^44 - 3 * q^46 - 5 * q^47 + 80 * q^49 + 33 * q^50 + 60 * q^52 - 9 * q^53 - 7 * q^55 - 44 * q^56 - 27 * q^58 - 13 * q^59 + 21 * q^61 + 23 * q^62 + 66 * q^64 - 25 * q^65 + 38 * q^67 + 74 * q^68 - 33 * q^70 - 12 * q^71 + 20 * q^73 + 12 * q^74 + 59 * q^76 + 5 * q^77 + q^79 + 38 * q^80 + 7 * q^82 + 19 * q^83 + 38 * q^85 + 3 * q^86 + 6 * q^88 - 37 * q^89 + 24 * q^91 - 31 * q^92 - 64 * q^94 + 43 * q^95 + 68 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

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Database

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