LMFDB - Embedded newform 6039.2.a.p.1.10

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:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	$q-0.608354 q^{2} -1.62991 q^{4} -1.59846 q^{5} -1.08777 q^{7} +2.20827 q^{8} +O(q^{10})$	\(q-0.608354 q^{2} -1.62991 q^{4} -1.59846 q^{5} -1.08777 q^{7} +2.20827 q^{8} +0.972431 q^{10} -1.00000 q^{11} +1.21304 q^{13} +0.661748 q^{14} +1.91640 q^{16} +1.83173 q^{17} +6.08763 q^{19} +2.60535 q^{20} +0.608354 q^{22} +8.06434 q^{23} -2.44491 q^{25} -0.737960 q^{26} +1.77296 q^{28} -4.42957 q^{29} -8.64234 q^{31} -5.58238 q^{32} -1.11434 q^{34} +1.73876 q^{35} +5.08068 q^{37} -3.70343 q^{38} -3.52983 q^{40} +9.86486 q^{41} -2.74478 q^{43} +1.62991 q^{44} -4.90597 q^{46} -8.57549 q^{47} -5.81676 q^{49} +1.48737 q^{50} -1.97715 q^{52} -1.82037 q^{53} +1.59846 q^{55} -2.40208 q^{56} +2.69475 q^{58} +3.94929 q^{59} +1.00000 q^{61} +5.25760 q^{62} -0.436746 q^{64} -1.93901 q^{65} -6.68123 q^{67} -2.98555 q^{68} -1.05778 q^{70} -15.1861 q^{71} +6.40289 q^{73} -3.09085 q^{74} -9.92227 q^{76} +1.08777 q^{77} +4.44589 q^{79} -3.06330 q^{80} -6.00132 q^{82} -2.59705 q^{83} -2.92795 q^{85} +1.66980 q^{86} -2.20827 q^{88} +12.4161 q^{89} -1.31951 q^{91} -13.1441 q^{92} +5.21693 q^{94} -9.73086 q^{95} +1.17398 q^{97} +3.53865 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) $ 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8	$ 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8} - 12 q^{10} - 25 q^{11} - 4 q^{13} + 14 q^{14} + 21 q^{16} + 16 q^{17} - 18 q^{19} + 28 q^{20} - 5 q^{22} + 8 q^{23} + 29 q^{25} + 16 q^{26} + 18 q^{28} + 28 q^{29} - 8 q^{31} + 35 q^{32} + 6 q^{34} + 22 q^{35} + 4 q^{37} - 4 q^{38} - 12 q^{40} + 58 q^{41} - 26 q^{43} - 25 q^{44} + 8 q^{46} + 20 q^{47} + 23 q^{49} + 27 q^{50} - 2 q^{52} + 36 q^{53} - 12 q^{55} + 70 q^{56} + 12 q^{58} + 18 q^{59} + 25 q^{61} + 42 q^{62} + 35 q^{64} + 76 q^{65} - 8 q^{67} + 28 q^{68} + 76 q^{70} + 24 q^{71} + 2 q^{73} + 40 q^{74} - 64 q^{76} + 4 q^{77} - 22 q^{79} + 36 q^{80} + 30 q^{82} + 14 q^{83} + 70 q^{86} - 15 q^{88} + 82 q^{89} - 6 q^{91} + 48 q^{92} - 16 q^{94} + 34 q^{95} + 16 q^{97} + 35 q^{98}+O(q^{100}) $ 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8 - 12 * q^10 - 25 * q^11 - 4 * q^13 + 14 * q^14 + 21 * q^16 + 16 * q^17 - 18 * q^19 + 28 * q^20 - 5 * q^22 + 8 * q^23 + 29 * q^25 + 16 * q^26 + 18 * q^28 + 28 * q^29 - 8 * q^31 + 35 * q^32 + 6 * q^34 + 22 * q^35 + 4 * q^37 - 4 * q^38 - 12 * q^40 + 58 * q^41 - 26 * q^43 - 25 * q^44 + 8 * q^46 + 20 * q^47 + 23 * q^49 + 27 * q^50 - 2 * q^52 + 36 * q^53 - 12 * q^55 + 70 * q^56 + 12 * q^58 + 18 * q^59 + 25 * q^61 + 42 * q^62 + 35 * q^64 + 76 * q^65 - 8 * q^67 + 28 * q^68 + 76 * q^70 + 24 * q^71 + 2 * q^73 + 40 * q^74 - 64 * q^76 + 4 * q^77 - 22 * q^79 + 36 * q^80 + 30 * q^82 + 14 * q^83 + 70 * q^86 - 15 * q^88 + 82 * q^89 - 6 * q^91 + 48 * q^92 - 16 * q^94 + 34 * q^95 + 16 * q^97 + 35 * 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 $

$2$	−0.608354	−0.430171	−0.215085	−	0.976595i	$-0.569003\pi$
$2$	−0.608354	−0.430171	−0.215085	+	0.976595i	$0.569003\pi$
$3$	0	0
$3$	0	0
$4$	−1.62991	−0.814953
$5$	−1.59846	−0.714855	−0.357427	−	0.933941i	$-0.616346\pi$
$5$	−1.59846	−0.714855	−0.357427	+	0.933941i	$0.616346\pi$
$6$	0	0
$7$	−1.08777	−0.411138	−0.205569	−	0.978643i	$-0.565905\pi$
$7$	−1.08777	−0.411138	−0.205569	+	0.978643i	$0.565905\pi$
$8$	2.20827	0.780740
$9$	0	0
$10$	0.972431	0.307510
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.21304	0.336438	0.168219	−	0.985750i	$-0.446198\pi$
$13$	1.21304	0.336438	0.168219	+	0.985750i	$0.446198\pi$
$14$	0.661748	0.176860
$15$	0	0
$16$	1.91640	0.479101
$17$	1.83173	0.444260	0.222130	−	0.975017i	$-0.428699\pi$
$17$	1.83173	0.444260	0.222130	+	0.975017i	$0.428699\pi$
$18$	0	0
$19$	6.08763	1.39660	0.698299	−	0.715806i	$-0.253940\pi$
$19$	6.08763	1.39660	0.698299	+	0.715806i	$0.253940\pi$
$20$	2.60535	0.582573
$21$	0	0
$22$	0.608354	0.129701
$23$	8.06434	1.68153	0.840765	−	0.541400i	$-0.182105\pi$
$23$	8.06434	1.68153	0.840765	+	0.541400i	$0.182105\pi$
$24$	0	0
$25$	−2.44491	−0.488983
$26$	−0.737960	−0.144726
$27$	0	0
$28$	1.77296	0.335058
$29$	−4.42957	−0.822551	−0.411275	−	0.911511i	$-0.634917\pi$
$29$	−4.42957	−0.822551	−0.411275	+	0.911511i	$0.634917\pi$
$30$	0	0
$31$	−8.64234	−1.55221	−0.776105	−	0.630604i	$-0.782807\pi$
$31$	−8.64234	−1.55221	−0.776105	+	0.630604i	$0.782807\pi$
$32$	−5.58238	−0.986836
$33$	0	0
$34$	−1.11434	−0.191108
$35$	1.73876	0.293904
$36$	0	0
$37$	5.08068	0.835258	0.417629	−	0.908618i	$-0.362861\pi$
$37$	5.08068	0.835258	0.417629	+	0.908618i	$0.362861\pi$
$38$	−3.70343	−0.600776
$39$	0	0
$40$	−3.52983	−0.558116
$41$	9.86486	1.54063	0.770316	−	0.637662i	$-0.220098\pi$
$41$	9.86486	1.54063	0.770316	+	0.637662i	$0.220098\pi$
$42$	0	0
$43$	−2.74478	−0.418575	−0.209287	−	0.977854i	$-0.567114\pi$
$43$	−2.74478	−0.418575	−0.209287	+	0.977854i	$0.567114\pi$
$44$	1.62991	0.245718
$45$	0	0
$46$	−4.90597	−0.723346
$47$	−8.57549	−1.25086	−0.625432	−	0.780279i	$-0.715077\pi$
$47$	−8.57549	−1.25086	−0.625432	+	0.780279i	$0.715077\pi$
$48$	0	0
$49$	−5.81676	−0.830965
$50$	1.48737	0.210346
$51$	0	0
$52$	−1.97715	−0.274181
$53$	−1.82037	−0.250047	−0.125023	−	0.992154i	$-0.539901\pi$
$53$	−1.82037	−0.250047	−0.125023	+	0.992154i	$0.539901\pi$
$54$	0	0
$55$	1.59846	0.215537
$56$	−2.40208	−0.320992
$57$	0	0
$58$	2.69475	0.353837
$59$	3.94929	0.514154	0.257077	−	0.966391i	$-0.417241\pi$
$59$	3.94929	0.514154	0.257077	+	0.966391i	$0.417241\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	5.25760	0.667716
$63$	0	0
$64$	−0.436746	−0.0545932
$65$	−1.93901	−0.240504
$66$	0	0
$67$	−6.68123	−0.816243	−0.408121	−	0.912928i	$-0.633816\pi$
$67$	−6.68123	−0.816243	−0.408121	+	0.912928i	$0.633816\pi$
$68$	−2.98555	−0.362051
$69$	0	0
$70$	−1.05778	−0.126429
$71$	−15.1861	−1.80226	−0.901129	−	0.433551i	$-0.857260\pi$
$71$	−15.1861	−1.80226	−0.901129	+	0.433551i	$0.857260\pi$
$72$	0	0
$73$	6.40289	0.749401	0.374701	−	0.927146i	$-0.377746\pi$
$73$	6.40289	0.749401	0.374701	+	0.927146i	$0.377746\pi$
$74$	−3.09085	−0.359304
$75$	0	0
$76$	−9.92227	−1.13816
$77$	1.08777	0.123963
$78$	0	0
$79$	4.44589	0.500201	0.250101	−	0.968220i	$-0.419536\pi$
$79$	4.44589	0.500201	0.250101	+	0.968220i	$0.419536\pi$
$80$	−3.06330	−0.342488
$81$	0	0
$82$	−6.00132	−0.662735
$83$	−2.59705	−0.285064	−0.142532	−	0.989790i	$-0.545524\pi$
$83$	−2.59705	−0.285064	−0.142532	+	0.989790i	$0.545524\pi$
$84$	0	0
$85$	−2.92795	−0.317581
$86$	1.66980	0.180059
$87$	0	0
$88$	−2.20827	−0.235402
$89$	12.4161	1.31611	0.658054	−	0.752971i	$-0.271380\pi$
$89$	12.4161	1.31611	0.658054	+	0.752971i	$0.271380\pi$
$90$	0	0
$91$	−1.31951	−0.138322
$92$	−13.1441	−1.37037
$93$	0	0
$94$	5.21693	0.538085
$95$	−9.73086	−0.998365
$96$	0	0
$97$	1.17398	0.119200	0.0595999	−	0.998222i	$-0.481018\pi$
$97$	1.17398	0.119200	0.0595999	+	0.998222i	$0.481018\pi$
$98$	3.53865	0.357457
$99$	0	0
$100$	3.98498	0.398498
$101$	−8.20938	−0.816864	−0.408432	−	0.912789i	$-0.633924\pi$
$101$	−8.20938	−0.816864	−0.408432	+	0.912789i	$0.633924\pi$
$102$	0	0
$103$	−11.6338	−1.14631	−0.573154	−	0.819448i	$-0.694280\pi$
$103$	−11.6338	−1.14631	−0.573154	+	0.819448i	$0.694280\pi$
$104$	2.67872	0.262671
$105$	0	0
$106$	1.10743	0.107563
$107$	6.48027	0.626471	0.313236	−	0.949675i	$-0.398587\pi$
$107$	6.48027	0.626471	0.313236	+	0.949675i	$0.398587\pi$
$108$	0	0
$109$	−14.4149	−1.38069	−0.690347	−	0.723479i	$-0.742542\pi$
$109$	−14.4149	−1.38069	−0.690347	+	0.723479i	$0.742542\pi$
$110$	−0.972431	−0.0927177
$111$	0	0
$112$	−2.08461	−0.196977
$113$	7.96917	0.749676	0.374838	−	0.927090i	$-0.377698\pi$
$113$	7.96917	0.749676	0.374838	+	0.927090i	$0.377698\pi$
$114$	0	0
$115$	−12.8906	−1.20205
$116$	7.21978	0.670340
$117$	0	0
$118$	−2.40257	−0.221174
$119$	−1.99250	−0.182652
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.608354	−0.0550777
$123$	0	0
$124$	14.0862	1.26498
$125$	11.9004	1.06441
$126$	0	0
$127$	−6.13560	−0.544447	−0.272223	−	0.962234i	$-0.587759\pi$
$127$	−6.13560	−0.544447	−0.272223	+	0.962234i	$0.587759\pi$
$128$	11.4305	1.01032
$129$	0	0
$130$	1.17960	0.103458
$131$	3.93457	0.343765	0.171882	−	0.985117i	$-0.445015\pi$
$131$	3.93457	0.343765	0.171882	+	0.985117i	$0.445015\pi$
$132$	0	0
$133$	−6.62194	−0.574195
$134$	4.06455	0.351124
$135$	0	0
$136$	4.04495	0.346851
$137$	1.99174	0.170166	0.0850830	−	0.996374i	$-0.472884\pi$
$137$	1.99174	0.170166	0.0850830	+	0.996374i	$0.472884\pi$
$138$	0	0
$139$	21.8129	1.85015	0.925073	−	0.379789i	$-0.124003\pi$
$139$	21.8129	1.85015	0.925073	+	0.379789i	$0.124003\pi$
$140$	−2.83401	−0.239518
$141$	0	0
$142$	9.23852	0.775279
$143$	−1.21304	−0.101440
$144$	0	0
$145$	7.08051	0.588004
$146$	−3.89522	−0.322371
$147$	0	0
$148$	−8.28103	−0.680696
$149$	8.98741	0.736278	0.368139	−	0.929771i	$-0.379995\pi$
$149$	8.98741	0.736278	0.368139	+	0.929771i	$0.379995\pi$
$150$	0	0
$151$	−15.3372	−1.24812	−0.624061	−	0.781376i	$-0.714518\pi$
$151$	−15.3372	−1.24812	−0.624061	+	0.781376i	$0.714518\pi$
$152$	13.4431	1.09038
$153$	0	0
$154$	−0.661748	−0.0533252
$155$	13.8145	1.10960
$156$	0	0
$157$	10.9878	0.876918	0.438459	−	0.898751i	$-0.355524\pi$
$157$	10.9878	0.876918	0.438459	+	0.898751i	$0.355524\pi$
$158$	−2.70467	−0.215172
$159$	0	0
$160$	8.92324	0.705444
$161$	−8.77214	−0.691341
$162$	0	0
$163$	14.2990	1.11999	0.559994	−	0.828497i	$-0.310804\pi$
$163$	14.2990	1.11999	0.559994	+	0.828497i	$0.310804\pi$
$164$	−16.0788	−1.25554
$165$	0	0
$166$	1.57993	0.122626
$167$	8.24474	0.637997	0.318999	−	0.947755i	$-0.396653\pi$
$167$	8.24474	0.637997	0.318999	+	0.947755i	$0.396653\pi$
$168$	0	0
$169$	−11.5285	−0.886810
$170$	1.78123	0.136614
$171$	0	0
$172$	4.47373	0.341119
$173$	20.3190	1.54482	0.772412	−	0.635121i	$-0.219050\pi$
$173$	20.3190	1.54482	0.772412	+	0.635121i	$0.219050\pi$
$174$	0	0
$175$	2.65950	0.201039
$176$	−1.91640	−0.144454
$177$	0	0
$178$	−7.55340	−0.566151
$179$	15.9678	1.19349	0.596744	−	0.802432i	$-0.296461\pi$
$179$	15.9678	1.19349	0.596744	+	0.802432i	$0.296461\pi$
$180$	0	0
$181$	−23.2460	−1.72786	−0.863932	−	0.503608i	$-0.832005\pi$
$181$	−23.2460	−1.72786	−0.863932	+	0.503608i	$0.832005\pi$
$182$	0.802730	0.0595023
$183$	0	0
$184$	17.8082	1.31284
$185$	−8.12128	−0.597088
$186$	0	0
$187$	−1.83173	−0.133949
$188$	13.9772	1.01939
$189$	0	0
$190$	5.91980	0.429468
$191$	15.9540	1.15439	0.577197	−	0.816605i	$-0.304147\pi$
$191$	15.9540	1.15439	0.577197	+	0.816605i	$0.304147\pi$
$192$	0	0
$193$	−8.86017	−0.637769	−0.318885	−	0.947794i	$-0.603308\pi$
$193$	−8.86017	−0.637769	−0.318885	+	0.947794i	$0.603308\pi$
$194$	−0.714196	−0.0512763
$195$	0	0
$196$	9.48077	0.677198
$197$	9.41388	0.670712	0.335356	−	0.942092i	$-0.391143\pi$
$197$	9.41388	0.670712	0.335356	+	0.942092i	$0.391143\pi$
$198$	0	0
$199$	10.4323	0.739525	0.369763	−	0.929126i	$-0.379439\pi$
$199$	10.4323	0.739525	0.369763	+	0.929126i	$0.379439\pi$
$200$	−5.39902	−0.381768
$201$	0	0
$202$	4.99421	0.351391
$203$	4.81835	0.338182
$204$	0	0
$205$	−15.7686	−1.10133
$206$	7.07744	0.493108
$207$	0	0
$208$	2.32468	0.161188
$209$	−6.08763	−0.421090
$210$	0	0
$211$	14.2344	0.979936	0.489968	−	0.871740i	$-0.337008\pi$
$211$	14.2344	0.979936	0.489968	+	0.871740i	$0.337008\pi$
$212$	2.96703	0.203777
$213$	0	0
$214$	−3.94230	−0.269490
$215$	4.38743	0.299220
$216$	0	0
$217$	9.40087	0.638173
$218$	8.76933	0.593934
$219$	0	0
$220$	−2.60535	−0.175652
$221$	2.22197	0.149466
$222$	0	0
$223$	−19.3856	−1.29816	−0.649079	−	0.760721i	$-0.724846\pi$
$223$	−19.3856	−1.29816	−0.649079	+	0.760721i	$0.724846\pi$
$224$	6.07235	0.405726
$225$	0	0
$226$	−4.84807	−0.322489
$227$	20.6868	1.37303	0.686517	−	0.727114i	$-0.259139\pi$
$227$	20.6868	1.37303	0.686517	+	0.727114i	$0.259139\pi$
$228$	0	0
$229$	−15.6682	−1.03538	−0.517692	−	0.855567i	$-0.673209\pi$
$229$	−15.6682	−1.03538	−0.517692	+	0.855567i	$0.673209\pi$
$230$	7.84201	0.517087
$231$	0	0
$232$	−9.78167	−0.642198
$233$	1.48548	0.0973172	0.0486586	−	0.998815i	$-0.484505\pi$
$233$	1.48548	0.0973172	0.0486586	+	0.998815i	$0.484505\pi$
$234$	0	0
$235$	13.7076	0.894186
$236$	−6.43698	−0.419012
$237$	0	0
$238$	1.21214	0.0785716
$239$	7.58548	0.490664	0.245332	−	0.969439i	$-0.421103\pi$
$239$	7.58548	0.490664	0.245332	+	0.969439i	$0.421103\pi$
$240$	0	0
$241$	19.4742	1.25444	0.627222	−	0.778840i	$-0.284192\pi$
$241$	19.4742	1.25444	0.627222	+	0.778840i	$0.284192\pi$
$242$	−0.608354	−0.0391065
$243$	0	0
$244$	−1.62991	−0.104344
$245$	9.29788	0.594020
$246$	0	0
$247$	7.38456	0.469869
$248$	−19.0846	−1.21187
$249$	0	0
$250$	−7.23967	−0.457877
$251$	−10.3379	−0.652521	−0.326261	−	0.945280i	$-0.605789\pi$
$251$	−10.3379	−0.652521	−0.326261	+	0.945280i	$0.605789\pi$
$252$	0	0
$253$	−8.06434	−0.507001
$254$	3.73262	0.234205
$255$	0	0
$256$	−6.08027	−0.380017
$257$	18.8670	1.17689	0.588445	−	0.808537i	$-0.299740\pi$
$257$	18.8670	1.17689	0.588445	+	0.808537i	$0.299740\pi$
$258$	0	0
$259$	−5.52661	−0.343407
$260$	3.16040	0.196000
$261$	0	0
$262$	−2.39361	−0.147878
$263$	−12.0888	−0.745429	−0.372714	−	0.927946i	$-0.621573\pi$
$263$	−12.0888	−0.745429	−0.372714	+	0.927946i	$0.621573\pi$
$264$	0	0
$265$	2.90979	0.178747
$266$	4.02848	0.247002
$267$	0	0
$268$	10.8898	0.665199
$269$	−5.15520	−0.314318	−0.157159	−	0.987573i	$-0.550234\pi$
$269$	−5.15520	−0.314318	−0.157159	+	0.987573i	$0.550234\pi$
$270$	0	0
$271$	−9.40705	−0.571438	−0.285719	−	0.958313i	$-0.592232\pi$
$271$	−9.40705	−0.571438	−0.285719	+	0.958313i	$0.592232\pi$
$272$	3.51034	0.212845
$273$	0	0
$274$	−1.21168	−0.0732005
$275$	2.44491	0.147434
$276$	0	0
$277$	−12.8365	−0.771269	−0.385634	−	0.922652i	$-0.626017\pi$
$277$	−12.8365	−0.771269	−0.385634	+	0.922652i	$0.626017\pi$
$278$	−13.2700	−0.795879
$279$	0	0
$280$	3.83964	0.229463
$281$	−2.48116	−0.148014	−0.0740068	−	0.997258i	$-0.523579\pi$
$281$	−2.48116	−0.148014	−0.0740068	+	0.997258i	$0.523579\pi$
$282$	0	0
$283$	4.86885	0.289423	0.144712	−	0.989474i	$-0.453775\pi$
$283$	4.86885	0.289423	0.144712	+	0.989474i	$0.453775\pi$
$284$	24.7519	1.46876
$285$	0	0
$286$	0.737960	0.0436365
$287$	−10.7307	−0.633413
$288$	0	0
$289$	−13.6448	−0.802633
$290$	−4.30745	−0.252942
$291$	0	0
$292$	−10.4361	−0.610727
$293$	6.33047	0.369830	0.184915	−	0.982755i	$-0.440799\pi$
$293$	6.33047	0.369830	0.184915	+	0.982755i	$0.440799\pi$
$294$	0	0
$295$	−6.31280	−0.367546
$296$	11.2195	0.652120
$297$	0	0
$298$	−5.46753	−0.316725
$299$	9.78240	0.565731
$300$	0	0
$301$	2.98569	0.172092
$302$	9.33042	0.536906
$303$	0	0
$304$	11.6664	0.669112
$305$	−1.59846	−0.0915278
$306$	0	0
$307$	17.4660	0.996839	0.498420	−	0.866936i	$-0.333914\pi$
$307$	17.4660	0.996839	0.498420	+	0.866936i	$0.333914\pi$
$308$	−1.77296	−0.101024
$309$	0	0
$310$	−8.40408	−0.477320
$311$	12.1001	0.686135	0.343068	−	0.939311i	$-0.388534\pi$
$311$	12.1001	0.686135	0.343068	+	0.939311i	$0.388534\pi$
$312$	0	0
$313$	−20.8514	−1.17859	−0.589294	−	0.807919i	$-0.700594\pi$
$313$	−20.8514	−1.17859	−0.589294	+	0.807919i	$0.700594\pi$
$314$	−6.68444	−0.377225
$315$	0	0
$316$	−7.24637	−0.407640
$317$	9.50752	0.533996	0.266998	−	0.963697i	$-0.413968\pi$
$317$	9.50752	0.533996	0.266998	+	0.963697i	$0.413968\pi$
$318$	0	0
$319$	4.42957	0.248008
$320$	0.698122	0.0390262
$321$	0	0
$322$	5.33656	0.297395
$323$	11.1509	0.620452
$324$	0	0
$325$	−2.96579	−0.164512
$326$	−8.69887	−0.481786
$327$	0	0
$328$	21.7842	1.20283
$329$	9.32815	0.514278
$330$	0	0
$331$	−0.591307	−0.0325012	−0.0162506	−	0.999868i	$-0.505173\pi$
$331$	−0.591307	−0.0325012	−0.0162506	+	0.999868i	$0.505173\pi$
$332$	4.23295	0.232314
$333$	0	0
$334$	−5.01572	−0.274448
$335$	10.6797	0.583495
$336$	0	0
$337$	−16.6177	−0.905224	−0.452612	−	0.891708i	$-0.649508\pi$
$337$	−16.6177	−0.905224	−0.452612	+	0.891708i	$0.649508\pi$
$338$	7.01342	0.381480
$339$	0	0
$340$	4.77229	0.258814
$341$	8.64234	0.468009
$342$	0	0
$343$	13.9417	0.752780
$344$	−6.06120	−0.326798
$345$	0	0
$346$	−12.3611	−0.664539
$347$	−22.8869	−1.22863	−0.614317	−	0.789059i	$-0.710569\pi$
$347$	−22.8869	−1.22863	−0.614317	+	0.789059i	$0.710569\pi$
$348$	0	0
$349$	4.96108	0.265561	0.132780	−	0.991145i	$-0.457610\pi$
$349$	4.96108	0.265561	0.132780	+	0.991145i	$0.457610\pi$
$350$	−1.61792	−0.0864813
$351$	0	0
$352$	5.58238	0.297542
$353$	−4.15688	−0.221249	−0.110624	−	0.993862i	$-0.535285\pi$
$353$	−4.15688	−0.221249	−0.110624	+	0.993862i	$0.535285\pi$
$354$	0	0
$355$	24.2744	1.28835
$356$	−20.2371	−1.07257
$357$	0	0
$358$	−9.71406	−0.513404
$359$	−0.180970	−0.00955121	−0.00477561	−	0.999989i	$-0.501520\pi$
$359$	−0.180970	−0.00955121	−0.00477561	+	0.999989i	$0.501520\pi$
$360$	0	0
$361$	18.0593	0.950487
$362$	14.1418	0.743277
$363$	0	0
$364$	2.15068	0.112726
$365$	−10.2348	−0.535713
$366$	0	0
$367$	36.6680	1.91406	0.957028	−	0.289996i	$-0.0936540\pi$
$367$	36.6680	1.91406	0.957028	+	0.289996i	$0.0936540\pi$
$368$	15.4545	0.805623
$369$	0	0
$370$	4.94061	0.256850
$371$	1.98014	0.102804
$372$	0	0
$373$	17.2408	0.892697	0.446348	−	0.894859i	$-0.352724\pi$
$373$	17.2408	0.892697	0.446348	+	0.894859i	$0.352724\pi$
$374$	1.11434	0.0576211
$375$	0	0
$376$	−18.9370	−0.976599
$377$	−5.37326	−0.276737
$378$	0	0
$379$	−11.1596	−0.573229	−0.286614	−	0.958046i	$-0.592530\pi$
$379$	−11.1596	−0.573229	−0.286614	+	0.958046i	$0.592530\pi$
$380$	15.8604	0.813621
$381$	0	0
$382$	−9.70569	−0.496586
$383$	1.93582	0.0989157	0.0494579	−	0.998776i	$-0.484251\pi$
$383$	1.93582	0.0989157	0.0494579	+	0.998776i	$0.484251\pi$
$384$	0	0
$385$	−1.73876	−0.0886154
$386$	5.39012	0.274350
$387$	0	0
$388$	−1.91348	−0.0971422
$389$	−34.5907	−1.75382	−0.876910	−	0.480655i	$-0.840399\pi$
$389$	−34.5907	−1.75382	−0.876910	+	0.480655i	$0.840399\pi$
$390$	0	0
$391$	14.7717	0.747036
$392$	−12.8450	−0.648768
$393$	0	0
$394$	−5.72697	−0.288521
$395$	−7.10659	−0.357571
$396$	0	0
$397$	11.6004	0.582205	0.291102	−	0.956692i	$-0.405978\pi$
$397$	11.6004	0.582205	0.291102	+	0.956692i	$0.405978\pi$
$398$	−6.34652	−0.318122
$399$	0	0
$400$	−4.68544	−0.234272
$401$	20.5659	1.02701	0.513506	−	0.858086i	$-0.328346\pi$
$401$	20.5659	1.02701	0.513506	+	0.858086i	$0.328346\pi$
$402$	0	0
$403$	−10.4835	−0.522222
$404$	13.3805	0.665706
$405$	0	0
$406$	−2.93126	−0.145476
$407$	−5.08068	−0.251840
$408$	0	0
$409$	33.5692	1.65989	0.829944	−	0.557847i	$-0.188372\pi$
$409$	33.5692	1.65989	0.829944	+	0.557847i	$0.188372\pi$
$410$	9.59290	0.473760
$411$	0	0
$412$	18.9619	0.934187
$413$	−4.29592	−0.211388
$414$	0	0
$415$	4.15130	0.203779
$416$	−6.77168	−0.332009
$417$	0	0
$418$	3.70343	0.181141
$419$	0.402780	0.0196771	0.00983854	−	0.999952i	$-0.496868\pi$
$419$	0.402780	0.0196771	0.00983854	+	0.999952i	$0.496868\pi$
$420$	0	0
$421$	−33.3080	−1.62333	−0.811666	−	0.584122i	$-0.801439\pi$
$421$	−33.3080	−1.62333	−0.811666	+	0.584122i	$0.801439\pi$
$422$	−8.65954	−0.421540
$423$	0	0
$424$	−4.01986	−0.195222
$425$	−4.47842	−0.217235
$426$	0	0
$427$	−1.08777	−0.0526408
$428$	−10.5622	−0.510545
$429$	0	0
$430$	−2.66911	−0.128716
$431$	32.7752	1.57873	0.789363	−	0.613927i	$-0.210411\pi$
$431$	32.7752	1.57873	0.789363	+	0.613927i	$0.210411\pi$
$432$	0	0
$433$	29.7947	1.43184	0.715921	−	0.698182i	$-0.246007\pi$
$433$	29.7947	1.43184	0.715921	+	0.698182i	$0.246007\pi$
$434$	−5.71905	−0.274523
$435$	0	0
$436$	23.4949	1.12520
$437$	49.0927	2.34842
$438$	0	0
$439$	1.58905	0.0758414	0.0379207	−	0.999281i	$-0.487927\pi$
$439$	1.58905	0.0758414	0.0379207	+	0.999281i	$0.487927\pi$
$440$	3.52983	0.168278
$441$	0	0
$442$	−1.35174	−0.0642958
$443$	3.58682	0.170415	0.0852075	−	0.996363i	$-0.472845\pi$
$443$	3.58682	0.170415	0.0852075	+	0.996363i	$0.472845\pi$
$444$	0	0
$445$	−19.8467	−0.940826
$446$	11.7933	0.558430
$447$	0	0
$448$	0.475079	0.0224454
$449$	2.49126	0.117570	0.0587850	−	0.998271i	$-0.481277\pi$
$449$	2.49126	0.117570	0.0587850	+	0.998271i	$0.481277\pi$
$450$	0	0
$451$	−9.86486	−0.464518
$452$	−12.9890	−0.610951
$453$	0	0
$454$	−12.5849	−0.590639
$455$	2.10919	0.0988804
$456$	0	0
$457$	39.1354	1.83068	0.915338	−	0.402687i	$-0.131924\pi$
$457$	39.1354	1.83068	0.915338	+	0.402687i	$0.131924\pi$
$458$	9.53181	0.445392
$459$	0	0
$460$	21.0104	0.979614
$461$	32.5404	1.51556	0.757778	−	0.652512i	$-0.226285\pi$
$461$	32.5404	1.51556	0.757778	+	0.652512i	$0.226285\pi$
$462$	0	0
$463$	24.2736	1.12809	0.564044	−	0.825744i	$-0.309245\pi$
$463$	24.2736	1.12809	0.564044	+	0.825744i	$0.309245\pi$
$464$	−8.48885	−0.394085
$465$	0	0
$466$	−0.903699	−0.0418630
$467$	−1.66725	−0.0771512	−0.0385756	−	0.999256i	$-0.512282\pi$
$467$	−1.66725	−0.0771512	−0.0385756	+	0.999256i	$0.512282\pi$
$468$	0	0
$469$	7.26764	0.335588
$470$	−8.33907	−0.384653
$471$	0	0
$472$	8.72110	0.401421
$473$	2.74478	0.126205
$474$	0	0
$475$	−14.8837	−0.682913
$476$	3.24759	0.148853
$477$	0	0
$478$	−4.61465	−0.211069
$479$	7.87827	0.359967	0.179984	−	0.983670i	$-0.442396\pi$
$479$	7.87827	0.359967	0.179984	+	0.983670i	$0.442396\pi$
$480$	0	0
$481$	6.16309	0.281013
$482$	−11.8472	−0.539625
$483$	0	0
$484$	−1.62991	−0.0740866
$485$	−1.87657	−0.0852105
$486$	0	0
$487$	14.9038	0.675356	0.337678	−	0.941262i	$-0.390358\pi$
$487$	14.9038	0.675356	0.337678	+	0.941262i	$0.390358\pi$
$488$	2.20827	0.0999635
$489$	0	0
$490$	−5.65640	−0.255530
$491$	13.8552	0.625279	0.312639	−	0.949872i	$-0.398787\pi$
$491$	13.8552	0.625279	0.312639	+	0.949872i	$0.398787\pi$
$492$	0	0
$493$	−8.11378	−0.365426
$494$	−4.49243	−0.202124
$495$	0	0
$496$	−16.5622	−0.743666
$497$	16.5190	0.740977
$498$	0	0
$499$	2.56571	0.114857	0.0574284	−	0.998350i	$-0.481710\pi$
$499$	2.56571	0.114857	0.0574284	+	0.998350i	$0.481710\pi$
$500$	−19.3966	−0.867441
$501$	0	0
$502$	6.28909	0.280696
$503$	42.6396	1.90121	0.950604	−	0.310407i	$-0.100465\pi$
$503$	42.6396	1.90121	0.950604	+	0.310407i	$0.100465\pi$
$504$	0	0
$505$	13.1224	0.583939
$506$	4.90597	0.218097
$507$	0	0
$508$	10.0005	0.443699
$509$	17.1054	0.758182	0.379091	−	0.925359i	$-0.376237\pi$
$509$	17.1054	0.758182	0.379091	+	0.925359i	$0.376237\pi$
$510$	0	0
$511$	−6.96486	−0.308107
$512$	−19.1620	−0.846848
$513$	0	0
$514$	−11.4778	−0.506264
$515$	18.5961	0.819444
$516$	0	0
$517$	8.57549	0.377149
$518$	3.36213	0.147724
$519$	0	0
$520$	−4.28184	−0.187771
$521$	−13.6707	−0.598925	−0.299463	−	0.954108i	$-0.596807\pi$
$521$	−13.6707	−0.598925	−0.299463	+	0.954108i	$0.596807\pi$
$522$	0	0
$523$	−41.4844	−1.81399	−0.906994	−	0.421144i	$-0.861629\pi$
$523$	−41.4844	−1.81399	−0.906994	+	0.421144i	$0.861629\pi$
$524$	−6.41297	−0.280152
$525$	0	0
$526$	7.35428	0.320662
$527$	−15.8304	−0.689584
$528$	0	0
$529$	42.0336	1.82755
$530$	−1.77018	−0.0768919
$531$	0	0
$532$	10.7931	0.467942
$533$	11.9665	0.518327
$534$	0	0
$535$	−10.3585	−0.447836
$536$	−14.7539	−0.637273
$537$	0	0
$538$	3.13618	0.135210
$539$	5.81676	0.250546
$540$	0	0
$541$	−4.63230	−0.199158	−0.0995791	−	0.995030i	$-0.531750\pi$
$541$	−4.63230	−0.199158	−0.0995791	+	0.995030i	$0.531750\pi$
$542$	5.72281	0.245816
$543$	0	0
$544$	−10.2254	−0.438411
$545$	23.0416	0.986995
$546$	0	0
$547$	19.5301	0.835045	0.417522	−	0.908667i	$-0.362899\pi$
$547$	19.5301	0.835045	0.417522	+	0.908667i	$0.362899\pi$
$548$	−3.24635	−0.138677
$549$	0	0
$550$	−1.48737	−0.0634218
$551$	−26.9656	−1.14877
$552$	0	0
$553$	−4.83610	−0.205652
$554$	7.80911	0.331777
$555$	0	0
$556$	−35.5530	−1.50778
$557$	3.85519	0.163350	0.0816748	−	0.996659i	$-0.473973\pi$
$557$	3.85519	0.163350	0.0816748	+	0.996659i	$0.473973\pi$
$558$	0	0
$559$	−3.32954	−0.140824
$560$	3.33217	0.140810
$561$	0	0
$562$	1.50942	0.0636712
$563$	−28.2000	−1.18849	−0.594244	−	0.804285i	$-0.702549\pi$
$563$	−28.2000	−1.18849	−0.594244	+	0.804285i	$0.702549\pi$
$564$	0	0
$565$	−12.7384	−0.535910
$566$	−2.96198	−0.124501
$567$	0	0
$568$	−33.5349	−1.40709
$569$	−22.7893	−0.955379	−0.477689	−	0.878529i	$-0.658526\pi$
$569$	−22.7893	−0.955379	−0.477689	+	0.878529i	$0.658526\pi$
$570$	0	0
$571$	−5.46815	−0.228835	−0.114418	−	0.993433i	$-0.536500\pi$
$571$	−5.46815	−0.228835	−0.114418	+	0.993433i	$0.536500\pi$
$572$	1.97715	0.0826687
$573$	0	0
$574$	6.52806	0.272476
$575$	−19.7166	−0.822240
$576$	0	0
$577$	12.8782	0.536125	0.268063	−	0.963401i	$-0.413617\pi$
$577$	12.8782	0.536125	0.268063	+	0.963401i	$0.413617\pi$
$578$	8.30084	0.345270
$579$	0	0
$580$	−11.5406	−0.479196
$581$	2.82500	0.117201
$582$	0	0
$583$	1.82037	0.0753920
$584$	14.1393	0.585087
$585$	0	0
$586$	−3.85117	−0.159090
$587$	−33.1012	−1.36623	−0.683117	−	0.730309i	$-0.739376\pi$
$587$	−33.1012	−1.36623	−0.683117	+	0.730309i	$0.739376\pi$
$588$	0	0
$589$	−52.6114	−2.16781
$590$	3.84042	0.158107
$591$	0	0
$592$	9.73664	0.400173
$593$	−15.2233	−0.625145	−0.312573	−	0.949894i	$-0.601191\pi$
$593$	−15.2233	−0.625145	−0.312573	+	0.949894i	$0.601191\pi$
$594$	0	0
$595$	3.18494	0.130570
$596$	−14.6486	−0.600032
$597$	0	0
$598$	−5.95116	−0.243361
$599$	13.9287	0.569112	0.284556	−	0.958659i	$-0.408154\pi$
$599$	13.9287	0.569112	0.284556	+	0.958659i	$0.408154\pi$
$600$	0	0
$601$	−29.8581	−1.21794	−0.608969	−	0.793194i	$-0.708417\pi$
$601$	−29.8581	−1.21794	−0.608969	+	0.793194i	$0.708417\pi$
$602$	−1.81635	−0.0740290
$603$	0	0
$604$	24.9981	1.01716
$605$	−1.59846	−0.0649868
$606$	0	0
$607$	−25.6269	−1.04016	−0.520082	−	0.854116i	$-0.674099\pi$
$607$	−25.6269	−1.04016	−0.520082	+	0.854116i	$0.674099\pi$
$608$	−33.9835	−1.37821
$609$	0	0
$610$	0.972431	0.0393726
$611$	−10.4024	−0.420838
$612$	0	0
$613$	17.6394	0.712447	0.356224	−	0.934401i	$-0.384064\pi$
$613$	17.6394	0.712447	0.356224	+	0.934401i	$0.384064\pi$
$614$	−10.6255	−0.428811
$615$	0	0
$616$	2.40208	0.0967827
$617$	1.02893	0.0414231	0.0207116	−	0.999785i	$-0.493407\pi$
$617$	1.02893	0.0414231	0.0207116	+	0.999785i	$0.493407\pi$
$618$	0	0
$619$	−22.4035	−0.900473	−0.450236	−	0.892909i	$-0.648660\pi$
$619$	−22.4035	−0.900473	−0.450236	+	0.892909i	$0.648660\pi$
$620$	−22.5163	−0.904276
$621$	0	0
$622$	−7.36116	−0.295155
$623$	−13.5059	−0.541102
$624$	0	0
$625$	−6.79783	−0.271913
$626$	12.6850	0.506994
$627$	0	0
$628$	−17.9090	−0.714647
$629$	9.30643	0.371072
$630$	0	0
$631$	1.74863	0.0696118	0.0348059	−	0.999394i	$-0.488919\pi$
$631$	1.74863	0.0696118	0.0348059	+	0.999394i	$0.488919\pi$
$632$	9.81770	0.390527
$633$	0	0
$634$	−5.78394	−0.229709
$635$	9.80754	0.389200
$636$	0	0
$637$	−7.05598	−0.279568
$638$	−2.69475	−0.106686
$639$	0	0
$640$	−18.2712	−0.722232
$641$	−9.58189	−0.378462	−0.189231	−	0.981933i	$-0.560599\pi$
$641$	−9.58189	−0.378462	−0.189231	+	0.981933i	$0.560599\pi$
$642$	0	0
$643$	16.0858	0.634363	0.317181	−	0.948365i	$-0.397264\pi$
$643$	16.0858	0.634363	0.317181	+	0.948365i	$0.397264\pi$
$644$	14.2978	0.563411
$645$	0	0
$646$	−6.78369	−0.266901
$647$	−23.6044	−0.927983	−0.463992	−	0.885840i	$-0.653583\pi$
$647$	−23.6044	−0.927983	−0.463992	+	0.885840i	$0.653583\pi$
$648$	0	0
$649$	−3.94929	−0.155023
$650$	1.80425	0.0707684
$651$	0	0
$652$	−23.3061	−0.912737
$653$	−27.9546	−1.09395	−0.546974	−	0.837150i	$-0.684220\pi$
$653$	−27.9546	−1.09395	−0.546974	+	0.837150i	$0.684220\pi$
$654$	0	0
$655$	−6.28926	−0.245742
$656$	18.9051	0.738119
$657$	0	0
$658$	−5.67481	−0.221227
$659$	9.25156	0.360390	0.180195	−	0.983631i	$-0.442327\pi$
$659$	9.25156	0.360390	0.180195	+	0.983631i	$0.442327\pi$
$660$	0	0
$661$	−24.0384	−0.934984	−0.467492	−	0.883997i	$-0.654842\pi$
$661$	−24.0384	−0.934984	−0.467492	+	0.883997i	$0.654842\pi$
$662$	0.359724	0.0139811
$663$	0	0
$664$	−5.73499	−0.222561
$665$	10.5849	0.410466
$666$	0	0
$667$	−35.7216	−1.38314
$668$	−13.4382	−0.519938
$669$	0	0
$670$	−6.49704	−0.251003
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	37.1748	1.43298	0.716491	−	0.697596i	$-0.245747\pi$
$673$	37.1748	1.43298	0.716491	+	0.697596i	$0.245747\pi$
$674$	10.1094	0.389401
$675$	0	0
$676$	18.7904	0.722708
$677$	44.0146	1.69162	0.845810	−	0.533484i	$-0.179118\pi$
$677$	44.0146	1.69162	0.845810	+	0.533484i	$0.179118\pi$
$678$	0	0
$679$	−1.27702	−0.0490075
$680$	−6.46570	−0.247948
$681$	0	0
$682$	−5.25760	−0.201324
$683$	10.8698	0.415920	0.207960	−	0.978137i	$-0.433318\pi$
$683$	10.8698	0.415920	0.207960	+	0.978137i	$0.433318\pi$
$684$	0	0
$685$	−3.18373	−0.121644
$686$	−8.48147	−0.323824
$687$	0	0
$688$	−5.26011	−0.200540
$689$	−2.20819	−0.0841253
$690$	0	0
$691$	2.11161	0.0803293	0.0401647	−	0.999193i	$-0.487212\pi$
$691$	2.11161	0.0803293	0.0401647	+	0.999193i	$0.487212\pi$
$692$	−33.1181	−1.25896
$693$	0	0
$694$	13.9233	0.528523
$695$	−34.8671	−1.32259
$696$	0	0
$697$	18.0698	0.684441
$698$	−3.01809	−0.114236
$699$	0	0
$700$	−4.33474	−0.163838
$701$	40.7240	1.53812	0.769062	−	0.639174i	$-0.220724\pi$
$701$	40.7240	1.53812	0.769062	+	0.639174i	$0.220724\pi$
$702$	0	0
$703$	30.9293	1.16652
$704$	0.436746	0.0164605
$705$	0	0
$706$	2.52886	0.0951747
$707$	8.92991	0.335844
$708$	0	0
$709$	5.49716	0.206450	0.103225	−	0.994658i	$-0.467084\pi$
$709$	5.49716	0.206450	0.103225	+	0.994658i	$0.467084\pi$
$710$	−14.7674	−0.554212
$711$	0	0
$712$	27.4181	1.02754
$713$	−69.6948	−2.61009
$714$	0	0
$715$	1.93901	0.0725147
$716$	−26.0260	−0.972637
$717$	0	0
$718$	0.110094	0.00410865
$719$	−9.95332	−0.371196	−0.185598	−	0.982626i	$-0.559422\pi$
$719$	−9.95332	−0.371196	−0.185598	+	0.982626i	$0.559422\pi$
$720$	0	0
$721$	12.6548	0.471291
$722$	−10.9864	−0.408872
$723$	0	0
$724$	37.8888	1.40813
$725$	10.8299	0.402213
$726$	0	0
$727$	−6.69396	−0.248265	−0.124133	−	0.992266i	$-0.539615\pi$
$727$	−6.69396	−0.248265	−0.124133	+	0.992266i	$0.539615\pi$
$728$	−2.91383	−0.107994
$729$	0	0
$730$	6.22637	0.230448
$731$	−5.02769	−0.185956
$732$	0	0
$733$	36.7962	1.35910	0.679549	−	0.733630i	$-0.262176\pi$
$733$	36.7962	1.35910	0.679549	+	0.733630i	$0.262176\pi$
$734$	−22.3071	−0.823371
$735$	0	0
$736$	−45.0182	−1.65939
$737$	6.68123	0.246106
$738$	0	0
$739$	44.6365	1.64198	0.820990	−	0.570942i	$-0.193422\pi$
$739$	44.6365	1.64198	0.820990	+	0.570942i	$0.193422\pi$
$740$	13.2369	0.486599
$741$	0	0
$742$	−1.20463	−0.0442232
$743$	32.1183	1.17831	0.589153	−	0.808022i	$-0.299462\pi$
$743$	32.1183	1.17831	0.589153	+	0.808022i	$0.299462\pi$
$744$	0	0
$745$	−14.3661	−0.526332
$746$	−10.4885	−0.384012
$747$	0	0
$748$	2.98555	0.109162
$749$	−7.04904	−0.257566
$750$	0	0
$751$	49.7522	1.81548	0.907741	−	0.419531i	$-0.137805\pi$
$751$	49.7522	1.81548	0.907741	+	0.419531i	$0.137805\pi$
$752$	−16.4341	−0.599290
$753$	0	0
$754$	3.26884	0.119044
$755$	24.5159	0.892225
$756$	0	0
$757$	−5.02634	−0.182686	−0.0913428	−	0.995820i	$-0.529116\pi$
$757$	−5.02634	−0.182686	−0.0913428	+	0.995820i	$0.529116\pi$
$758$	6.78897	0.246586
$759$	0	0
$760$	−21.4883	−0.779464
$761$	−2.37697	−0.0861651	−0.0430825	−	0.999072i	$-0.513718\pi$
$761$	−2.37697	−0.0861651	−0.0430825	+	0.999072i	$0.513718\pi$
$762$	0	0
$763$	15.6800	0.567656
$764$	−26.0036	−0.940776
$765$	0	0
$766$	−1.17766	−0.0425507
$767$	4.79067	0.172981
$768$	0	0
$769$	−6.34114	−0.228667	−0.114334	−	0.993442i	$-0.536473\pi$
$769$	−6.34114	−0.228667	−0.114334	+	0.993442i	$0.536473\pi$
$770$	1.05778	0.0381198
$771$	0	0
$772$	14.4412	0.519752
$773$	−16.5070	−0.593716	−0.296858	−	0.954922i	$-0.595939\pi$
$773$	−16.5070	−0.593716	−0.296858	+	0.954922i	$0.595939\pi$
$774$	0	0
$775$	21.1298	0.759004
$776$	2.59246	0.0930640
$777$	0	0
$778$	21.0434	0.754442
$779$	60.0536	2.15165
$780$	0	0
$781$	15.1861	0.543401
$782$	−8.98641	−0.321353
$783$	0	0
$784$	−11.1473	−0.398117
$785$	−17.5635	−0.626869
$786$	0	0
$787$	−24.4820	−0.872688	−0.436344	−	0.899780i	$-0.643727\pi$
$787$	−24.4820	−0.872688	−0.436344	+	0.899780i	$0.643727\pi$
$788$	−15.3437	−0.546598
$789$	0	0
$790$	4.32332	0.153817
$791$	−8.66862	−0.308221
$792$	0	0
$793$	1.21304	0.0430765
$794$	−7.05712	−0.250448
$795$	0	0
$796$	−17.0036	−0.602678
$797$	35.3876	1.25349	0.626747	−	0.779223i	$-0.284386\pi$
$797$	35.3876	1.25349	0.626747	+	0.779223i	$0.284386\pi$
$798$	0	0
$799$	−15.7080	−0.555708
$800$	13.6484	0.482546
$801$	0	0
$802$	−12.5113	−0.441791
$803$	−6.40289	−0.225953
$804$	0	0
$805$	14.0219	0.494209
$806$	6.37770	0.224645
$807$	0	0
$808$	−18.1285	−0.637758
$809$	17.6287	0.619792	0.309896	−	0.950770i	$-0.399706\pi$
$809$	17.6287	0.619792	0.309896	+	0.950770i	$0.399706\pi$
$810$	0	0
$811$	−35.8352	−1.25835	−0.629173	−	0.777266i	$-0.716606\pi$
$811$	−35.8352	−1.25835	−0.629173	+	0.777266i	$0.716606\pi$
$812$	−7.85346	−0.275602
$813$	0	0
$814$	3.09085	0.108334
$815$	−22.8565	−0.800628
$816$	0	0
$817$	−16.7092	−0.584581
$818$	−20.4219	−0.714035
$819$	0	0
$820$	25.7014	0.897531
$821$	35.0933	1.22476	0.612382	−	0.790562i	$-0.290211\pi$
$821$	35.0933	1.22476	0.612382	+	0.790562i	$0.290211\pi$
$822$	0	0
$823$	21.6045	0.753085	0.376542	−	0.926399i	$-0.377113\pi$
$823$	21.6045	0.753085	0.376542	+	0.926399i	$0.377113\pi$
$824$	−25.6904	−0.894968
$825$	0	0
$826$	2.61344	0.0909332
$827$	33.1244	1.15185	0.575925	−	0.817503i	$-0.304642\pi$
$827$	33.1244	1.15185	0.575925	+	0.817503i	$0.304642\pi$
$828$	0	0
$829$	49.6496	1.72440	0.862201	−	0.506566i	$-0.169085\pi$
$829$	49.6496	1.72440	0.862201	+	0.506566i	$0.169085\pi$
$830$	−2.52546	−0.0876599
$831$	0	0
$832$	−0.529792	−0.0183672
$833$	−10.6547	−0.369164
$834$	0	0
$835$	−13.1789	−0.456075
$836$	9.92227	0.343169
$837$	0	0
$838$	−0.245032	−0.00846451
$839$	−3.42206	−0.118143	−0.0590713	−	0.998254i	$-0.518814\pi$
$839$	−3.42206	−0.118143	−0.0590713	+	0.998254i	$0.518814\pi$
$840$	0	0
$841$	−9.37890	−0.323410
$842$	20.2630	0.698310
$843$	0	0
$844$	−23.2007	−0.798602
$845$	18.4279	0.633940
$846$	0	0
$847$	−1.08777	−0.0373762
$848$	−3.48856	−0.119798
$849$	0	0
$850$	2.72446	0.0934483
$851$	40.9723	1.40451
$852$	0	0
$853$	−10.2579	−0.351223	−0.175612	−	0.984460i	$-0.556190\pi$
$853$	−10.2579	−0.351223	−0.175612	+	0.984460i	$0.556190\pi$
$854$	0.661748	0.0226446
$855$	0	0
$856$	14.3102	0.489111
$857$	6.70600	0.229073	0.114536	−	0.993419i	$-0.463462\pi$
$857$	6.70600	0.229073	0.114536	+	0.993419i	$0.463462\pi$
$858$	0	0
$859$	28.8911	0.985753	0.492877	−	0.870099i	$-0.335945\pi$
$859$	28.8911	0.985753	0.492877	+	0.870099i	$0.335945\pi$
$860$	−7.15110	−0.243850
$861$	0	0
$862$	−19.9389	−0.679122
$863$	−32.8812	−1.11929	−0.559644	−	0.828733i	$-0.689062\pi$
$863$	−32.8812	−1.11929	−0.559644	+	0.828733i	$0.689062\pi$
$864$	0	0
$865$	−32.4792	−1.10433
$866$	−18.1257	−0.615937
$867$	0	0
$868$	−15.3225	−0.520081
$869$	−4.44589	−0.150816
$870$	0	0
$871$	−8.10463	−0.274615
$872$	−31.8319	−1.07796
$873$	0	0
$874$	−29.8657	−1.01022
$875$	−12.9449	−0.437618
$876$	0	0
$877$	−25.9931	−0.877724	−0.438862	−	0.898555i	$-0.644618\pi$
$877$	−25.9931	−0.877724	−0.438862	+	0.898555i	$0.644618\pi$
$878$	−0.966706	−0.0326248
$879$	0	0
$880$	3.06330	0.103264
$881$	35.5316	1.19709	0.598545	−	0.801089i	$-0.295746\pi$
$881$	35.5316	1.19709	0.598545	+	0.801089i	$0.295746\pi$
$882$	0	0
$883$	15.2619	0.513605	0.256802	−	0.966464i	$-0.417331\pi$
$883$	15.2619	0.513605	0.256802	+	0.966464i	$0.417331\pi$
$884$	−3.62160	−0.121808
$885$	0	0
$886$	−2.18206	−0.0733076
$887$	8.55636	0.287294	0.143647	−	0.989629i	$-0.454117\pi$
$887$	8.55636	0.287294	0.143647	+	0.989629i	$0.454117\pi$
$888$	0	0
$889$	6.67412	0.223843
$890$	12.0738	0.404716
$891$	0	0
$892$	31.5968	1.05794
$893$	−52.2044	−1.74695
$894$	0	0
$895$	−25.5239	−0.853171
$896$	−12.4337	−0.415381
$897$	0	0
$898$	−1.51557	−0.0505752
$899$	38.2819	1.27677
$900$	0	0
$901$	−3.33442	−0.111086
$902$	6.00132	0.199822
$903$	0	0
$904$	17.5981	0.585302
$905$	37.1579	1.23517
$906$	0	0
$907$	−9.19951	−0.305465	−0.152732	−	0.988268i	$-0.548807\pi$
$907$	−9.19951	−0.305465	−0.152732	+	0.988268i	$0.548807\pi$
$908$	−33.7176	−1.11896
$909$	0	0
$910$	−1.28313	−0.0425355
$911$	−29.5429	−0.978802	−0.489401	−	0.872059i	$-0.662784\pi$
$911$	−29.5429	−0.978802	−0.489401	+	0.872059i	$0.662784\pi$
$912$	0	0
$913$	2.59705	0.0859500
$914$	−23.8081	−0.787503
$915$	0	0
$916$	25.5377	0.843790
$917$	−4.27990	−0.141335
$918$	0	0
$919$	−28.4118	−0.937220	−0.468610	−	0.883405i	$-0.655245\pi$
$919$	−28.4118	−0.937220	−0.468610	+	0.883405i	$0.655245\pi$
$920$	−28.4658	−0.938489
$921$	0	0
$922$	−19.7961	−0.651949
$923$	−18.4214	−0.606348
$924$	0	0
$925$	−12.4218	−0.408427
$926$	−14.7669	−0.485271
$927$	0	0
$928$	24.7276	0.811722
$929$	6.69706	0.219724	0.109862	−	0.993947i	$-0.464959\pi$
$929$	6.69706	0.219724	0.109862	+	0.993947i	$0.464959\pi$
$930$	0	0
$931$	−35.4103	−1.16053
$932$	−2.42120	−0.0793090
$933$	0	0
$934$	1.01428	0.0331882
$935$	2.92795	0.0957543
$936$	0	0
$937$	1.30770	0.0427208	0.0213604	−	0.999772i	$-0.493200\pi$
$937$	1.30770	0.0427208	0.0213604	+	0.999772i	$0.493200\pi$
$938$	−4.42129	−0.144360
$939$	0	0
$940$	−22.3421	−0.728719
$941$	−49.5909	−1.61662	−0.808308	−	0.588759i	$-0.799617\pi$
$941$	−49.5909	−1.61662	−0.808308	+	0.588759i	$0.799617\pi$
$942$	0	0
$943$	79.5536	2.59062
$944$	7.56845	0.246332
$945$	0	0
$946$	−1.66980	−0.0542898
$947$	−50.8509	−1.65243	−0.826216	−	0.563353i	$-0.809511\pi$
$947$	−50.8509	−1.65243	−0.826216	+	0.563353i	$0.809511\pi$
$948$	0	0
$949$	7.76698	0.252127
$950$	9.05457	0.293769
$951$	0	0
$952$	−4.39997	−0.142604
$953$	1.53393	0.0496887	0.0248444	−	0.999691i	$-0.492091\pi$
$953$	1.53393	0.0496887	0.0248444	+	0.999691i	$0.492091\pi$
$954$	0	0
$955$	−25.5019	−0.825223
$956$	−12.3636	−0.399868
$957$	0	0
$958$	−4.79277	−0.154847
$959$	−2.16656	−0.0699618
$960$	0	0
$961$	43.6901	1.40936
$962$	−3.74934	−0.120883
$963$	0	0
$964$	−31.7411	−1.02231
$965$	14.1627	0.455912
$966$	0	0
$967$	−23.8016	−0.765408	−0.382704	−	0.923871i	$-0.625007\pi$
$967$	−23.8016	−0.765408	−0.382704	+	0.923871i	$0.625007\pi$
$968$	2.20827	0.0709764
$969$	0	0
$970$	1.14162	0.0366551
$971$	50.2998	1.61420	0.807098	−	0.590417i	$-0.201037\pi$
$971$	50.2998	1.61420	0.807098	+	0.590417i	$0.201037\pi$
$972$	0	0
$973$	−23.7274	−0.760666
$974$	−9.06679	−0.290519
$975$	0	0
$976$	1.91640	0.0613426
$977$	20.0047	0.640008	0.320004	−	0.947416i	$-0.396316\pi$
$977$	20.0047	0.640008	0.320004	+	0.947416i	$0.396316\pi$
$978$	0	0
$979$	−12.4161	−0.396821
$980$	−15.1547	−0.484098
$981$	0	0
$982$	−8.42889	−0.268977
$983$	−31.6288	−1.00880	−0.504401	−	0.863470i	$-0.668287\pi$
$983$	−31.6288	−1.00880	−0.504401	+	0.863470i	$0.668287\pi$
$984$	0	0
$985$	−15.0478	−0.479461
$986$	4.93604	0.157196
$987$	0	0
$988$	−12.0361	−0.382921
$989$	−22.1348	−0.703847
$990$	0	0
$991$	36.5715	1.16173	0.580867	−	0.813999i	$-0.302714\pi$
$991$	36.5715	1.16173	0.580867	+	0.813999i	$0.302714\pi$
$992$	48.2449	1.53178
$993$	0	0
$994$	−10.0494	−0.318747
$995$	−16.6756	−0.528653
$996$	0	0
$997$	37.6392	1.19204	0.596022	−	0.802968i	$-0.296747\pi$
$997$	37.6392	1.19204	0.596022	+	0.802968i	$0.296747\pi$
$998$	−1.56086	−0.0494080
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.p.1.10	yes	25
3.2	odd	2		6039.2.a.m.1.16	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.16	✓	25	3.2	odd	2
6039.2.a.p.1.10	yes	25	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-0.608354 q^{2} -1.62991 q^{4} -1.59846 q^{5} -1.08777 q^{7} +2.20827 q^{8} +O(q^{10})\)	\(q-0.608354 q^{2} -1.62991 q^{4} -1.59846 q^{5} -1.08777 q^{7} +2.20827 q^{8} +0.972431 q^{10} -1.00000 q^{11} +1.21304 q^{13} +0.661748 q^{14} +1.91640 q^{16} +1.83173 q^{17} +6.08763 q^{19} +2.60535 q^{20} +0.608354 q^{22} +8.06434 q^{23} -2.44491 q^{25} -0.737960 q^{26} +1.77296 q^{28} -4.42957 q^{29} -8.64234 q^{31} -5.58238 q^{32} -1.11434 q^{34} +1.73876 q^{35} +5.08068 q^{37} -3.70343 q^{38} -3.52983 q^{40} +9.86486 q^{41} -2.74478 q^{43} +1.62991 q^{44} -4.90597 q^{46} -8.57549 q^{47} -5.81676 q^{49} +1.48737 q^{50} -1.97715 q^{52} -1.82037 q^{53} +1.59846 q^{55} -2.40208 q^{56} +2.69475 q^{58} +3.94929 q^{59} +1.00000 q^{61} +5.25760 q^{62} -0.436746 q^{64} -1.93901 q^{65} -6.68123 q^{67} -2.98555 q^{68} -1.05778 q^{70} -15.1861 q^{71} +6.40289 q^{73} -3.09085 q^{74} -9.92227 q^{76} +1.08777 q^{77} +4.44589 q^{79} -3.06330 q^{80} -6.00132 q^{82} -2.59705 q^{83} -2.92795 q^{85} +1.66980 q^{86} -2.20827 q^{88} +12.4161 q^{89} -1.31951 q^{91} -13.1441 q^{92} +5.21693 q^{94} -9.73086 q^{95} +1.17398 q^{97} +3.53865 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) \) 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8} - 12 q^{10} - 25 q^{11} - 4 q^{13} + 14 q^{14} + 21 q^{16} + 16 q^{17} - 18 q^{19} + 28 q^{20} - 5 q^{22} + 8 q^{23} + 29 q^{25} + 16 q^{26} + 18 q^{28} + 28 q^{29} - 8 q^{31} + 35 q^{32} + 6 q^{34} + 22 q^{35} + 4 q^{37} - 4 q^{38} - 12 q^{40} + 58 q^{41} - 26 q^{43} - 25 q^{44} + 8 q^{46} + 20 q^{47} + 23 q^{49} + 27 q^{50} - 2 q^{52} + 36 q^{53} - 12 q^{55} + 70 q^{56} + 12 q^{58} + 18 q^{59} + 25 q^{61} + 42 q^{62} + 35 q^{64} + 76 q^{65} - 8 q^{67} + 28 q^{68} + 76 q^{70} + 24 q^{71} + 2 q^{73} + 40 q^{74} - 64 q^{76} + 4 q^{77} - 22 q^{79} + 36 q^{80} + 30 q^{82} + 14 q^{83} + 70 q^{86} - 15 q^{88} + 82 q^{89} - 6 q^{91} + 48 q^{92} - 16 q^{94} + 34 q^{95} + 16 q^{97} + 35 q^{98}+O(q^{100}) \) 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8 - 12 * q^10 - 25 * q^11 - 4 * q^13 + 14 * q^14 + 21 * q^16 + 16 * q^17 - 18 * q^19 + 28 * q^20 - 5 * q^22 + 8 * q^23 + 29 * q^25 + 16 * q^26 + 18 * q^28 + 28 * q^29 - 8 * q^31 + 35 * q^32 + 6 * q^34 + 22 * q^35 + 4 * q^37 - 4 * q^38 - 12 * q^40 + 58 * q^41 - 26 * q^43 - 25 * q^44 + 8 * q^46 + 20 * q^47 + 23 * q^49 + 27 * q^50 - 2 * q^52 + 36 * q^53 - 12 * q^55 + 70 * q^56 + 12 * q^58 + 18 * q^59 + 25 * q^61 + 42 * q^62 + 35 * q^64 + 76 * q^65 - 8 * q^67 + 28 * q^68 + 76 * q^70 + 24 * q^71 + 2 * q^73 + 40 * q^74 - 64 * q^76 + 4 * q^77 - 22 * q^79 + 36 * q^80 + 30 * q^82 + 14 * q^83 + 70 * q^86 - 15 * q^88 + 82 * q^89 - 6 * q^91 + 48 * q^92 - 16 * q^94 + 34 * q^95 + 16 * q^97 + 35 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more