LMFDB - Embedded newform 6039.2.a.i.1.8

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} + \cdots - 47 $ x^13 - 2x^12 - 19x^11 + 35x^10 + 136x^9 - 220x^8 - 469x^7 + 610x^6 + 841x^5 - 760x^4 - 742x^3 + 366x^2 + 236x - 47
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.805107$ of defining polynomial
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)

$f(q)$	$=$	$q+0.805107 q^{2} -1.35180 q^{4} -1.06503 q^{5} -0.203035 q^{7} -2.69856 q^{8} +O(q^{10})$	\(q+0.805107 q^{2} -1.35180 q^{4} -1.06503 q^{5} -0.203035 q^{7} -2.69856 q^{8} -0.857467 q^{10} -1.00000 q^{11} +0.801491 q^{13} -0.163465 q^{14} +0.530979 q^{16} +0.376525 q^{17} +2.62261 q^{19} +1.43972 q^{20} -0.805107 q^{22} -2.93099 q^{23} -3.86570 q^{25} +0.645286 q^{26} +0.274464 q^{28} -0.907071 q^{29} +2.07822 q^{31} +5.82461 q^{32} +0.303143 q^{34} +0.216240 q^{35} -2.14816 q^{37} +2.11148 q^{38} +2.87406 q^{40} +1.83276 q^{41} -2.25276 q^{43} +1.35180 q^{44} -2.35976 q^{46} +0.685433 q^{47} -6.95878 q^{49} -3.11230 q^{50} -1.08346 q^{52} -14.0454 q^{53} +1.06503 q^{55} +0.547903 q^{56} -0.730289 q^{58} -7.73715 q^{59} -1.00000 q^{61} +1.67319 q^{62} +3.62748 q^{64} -0.853616 q^{65} -7.59429 q^{67} -0.508987 q^{68} +0.174096 q^{70} +9.23954 q^{71} +7.49398 q^{73} -1.72950 q^{74} -3.54525 q^{76} +0.203035 q^{77} +7.41703 q^{79} -0.565511 q^{80} +1.47557 q^{82} +8.55136 q^{83} -0.401012 q^{85} -1.81371 q^{86} +2.69856 q^{88} +2.97849 q^{89} -0.162731 q^{91} +3.96212 q^{92} +0.551846 q^{94} -2.79317 q^{95} +17.8193 q^{97} -5.60256 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * 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 $

$2$	0.805107	0.569296	0.284648	−	0.958632i	$-0.408123\pi$
$2$	0.805107	0.569296	0.284648	+	0.958632i	$0.408123\pi$
$3$	0	0
$3$	0	0
$4$	−1.35180	−0.675902
$5$	−1.06503	−0.476298	−0.238149	−	0.971229i	$-0.576541\pi$
$5$	−1.06503	−0.476298	−0.238149	+	0.971229i	$0.576541\pi$
$6$	0	0
$7$	−0.203035	−0.0767401	−0.0383701	−	0.999264i	$-0.512217\pi$
$7$	−0.203035	−0.0767401	−0.0383701	+	0.999264i	$0.512217\pi$
$8$	−2.69856	−0.954085
$9$	0	0
$10$	−0.857467	−0.271155
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.801491	0.222294	0.111147	−	0.993804i	$-0.464548\pi$
$13$	0.801491	0.222294	0.111147	+	0.993804i	$0.464548\pi$
$14$	−0.163465	−0.0436879
$15$	0	0
$16$	0.530979	0.132745
$17$	0.376525	0.0913207	0.0456603	−	0.998957i	$-0.485461\pi$
$17$	0.376525	0.0913207	0.0456603	+	0.998957i	$0.485461\pi$
$18$	0	0
$19$	2.62261	0.601667	0.300833	−	0.953677i	$-0.402735\pi$
$19$	2.62261	0.601667	0.300833	+	0.953677i	$0.402735\pi$
$20$	1.43972	0.321931
$21$	0	0
$22$	−0.805107	−0.171649
$23$	−2.93099	−0.611154	−0.305577	−	0.952167i	$-0.598849\pi$
$23$	−2.93099	−0.611154	−0.305577	+	0.952167i	$0.598849\pi$
$24$	0	0
$25$	−3.86570	−0.773140
$26$	0.645286	0.126551
$27$	0	0
$28$	0.274464	0.0518688
$29$	−0.907071	−0.168439	−0.0842195	−	0.996447i	$-0.526840\pi$
$29$	−0.907071	−0.168439	−0.0842195	+	0.996447i	$0.526840\pi$
$30$	0	0
$31$	2.07822	0.373260	0.186630	−	0.982430i	$-0.440243\pi$
$31$	2.07822	0.373260	0.186630	+	0.982430i	$0.440243\pi$
$32$	5.82461	1.02966
$33$	0	0
$34$	0.303143	0.0519885
$35$	0.216240	0.0365512
$36$	0	0
$37$	−2.14816	−0.353155	−0.176578	−	0.984287i	$-0.556503\pi$
$37$	−2.14816	−0.353155	−0.176578	+	0.984287i	$0.556503\pi$
$38$	2.11148	0.342527
$39$	0	0
$40$	2.87406	0.454429
$41$	1.83276	0.286229	0.143115	−	0.989706i	$-0.454288\pi$
$41$	1.83276	0.286229	0.143115	+	0.989706i	$0.454288\pi$
$42$	0	0
$43$	−2.25276	−0.343542	−0.171771	−	0.985137i	$-0.554949\pi$
$43$	−2.25276	−0.343542	−0.171771	+	0.985137i	$0.554949\pi$
$44$	1.35180	0.203792
$45$	0	0
$46$	−2.35976	−0.347928
$47$	0.685433	0.0999806	0.0499903	−	0.998750i	$-0.484081\pi$
$47$	0.685433	0.0999806	0.0499903	+	0.998750i	$0.484081\pi$
$48$	0	0
$49$	−6.95878	−0.994111
$50$	−3.11230	−0.440146
$51$	0	0
$52$	−1.08346	−0.150249
$53$	−14.0454	−1.92928	−0.964640	−	0.263570i	$-0.915100\pi$
$53$	−14.0454	−1.92928	−0.964640	+	0.263570i	$0.915100\pi$
$54$	0	0
$55$	1.06503	0.143609
$56$	0.547903	0.0732166
$57$	0	0
$58$	−0.730289	−0.0958917
$59$	−7.73715	−1.00729	−0.503646	−	0.863910i	$-0.668008\pi$
$59$	−7.73715	−1.00729	−0.503646	+	0.863910i	$0.668008\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	1.67319	0.212495
$63$	0	0
$64$	3.62748	0.453435
$65$	−0.853616	−0.105878
$66$	0	0
$67$	−7.59429	−0.927790	−0.463895	−	0.885890i	$-0.653549\pi$
$67$	−7.59429	−0.927790	−0.463895	+	0.885890i	$0.653549\pi$
$68$	−0.508987	−0.0617238
$69$	0	0
$70$	0.174096	0.0208085
$71$	9.23954	1.09653	0.548266	−	0.836304i	$-0.315288\pi$
$71$	9.23954	1.09653	0.548266	+	0.836304i	$0.315288\pi$
$72$	0	0
$73$	7.49398	0.877104	0.438552	−	0.898706i	$-0.355491\pi$
$73$	7.49398	0.877104	0.438552	+	0.898706i	$0.355491\pi$
$74$	−1.72950	−0.201050
$75$	0	0
$76$	−3.54525	−0.406668
$77$	0.203035	0.0231380
$78$	0	0
$79$	7.41703	0.834481	0.417240	−	0.908796i	$-0.362997\pi$
$79$	7.41703	0.834481	0.417240	+	0.908796i	$0.362997\pi$
$80$	−0.565511	−0.0632260
$81$	0	0
$82$	1.47557	0.162949
$83$	8.55136	0.938634	0.469317	−	0.883030i	$-0.344500\pi$
$83$	8.55136	0.938634	0.469317	+	0.883030i	$0.344500\pi$
$84$	0	0
$85$	−0.401012	−0.0434959
$86$	−1.81371	−0.195577
$87$	0	0
$88$	2.69856	0.287667
$89$	2.97849	0.315719	0.157860	−	0.987462i	$-0.449541\pi$
$89$	2.97849	0.315719	0.157860	+	0.987462i	$0.449541\pi$
$90$	0	0
$91$	−0.162731	−0.0170588
$92$	3.96212	0.413080
$93$	0	0
$94$	0.551846	0.0569186
$95$	−2.79317	−0.286573
$96$	0	0
$97$	17.8193	1.80928	0.904638	−	0.426181i	$-0.140141\pi$
$97$	17.8193	1.80928	0.904638	+	0.426181i	$0.140141\pi$
$98$	−5.60256	−0.565944
$99$	0	0
$100$	5.22567	0.522567
$101$	4.50884	0.448646	0.224323	−	0.974515i	$-0.427983\pi$
$101$	4.50884	0.448646	0.224323	+	0.974515i	$0.427983\pi$
$102$	0	0
$103$	19.8815	1.95898	0.979489	−	0.201499i	$-0.0645812\pi$
$103$	19.8815	1.95898	0.979489	+	0.201499i	$0.0645812\pi$
$104$	−2.16287	−0.212087
$105$	0	0
$106$	−11.3080	−1.09833
$107$	−6.58861	−0.636945	−0.318472	−	0.947932i	$-0.603170\pi$
$107$	−6.58861	−0.636945	−0.318472	+	0.947932i	$0.603170\pi$
$108$	0	0
$109$	19.1473	1.83398	0.916990	−	0.398910i	$-0.130612\pi$
$109$	19.1473	1.83398	0.916990	+	0.398910i	$0.130612\pi$
$110$	0.857467	0.0817562
$111$	0	0
$112$	−0.107807	−0.0101868
$113$	2.84072	0.267232	0.133616	−	0.991033i	$-0.457341\pi$
$113$	2.84072	0.267232	0.133616	+	0.991033i	$0.457341\pi$
$114$	0	0
$115$	3.12161	0.291091
$116$	1.22618	0.113848
$117$	0	0
$118$	−6.22923	−0.573447
$119$	−0.0764478	−0.00700796
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.805107	−0.0728909
$123$	0	0
$124$	−2.80935	−0.252287
$125$	9.44228	0.844543
$126$	0	0
$127$	5.00348	0.443987	0.221993	−	0.975048i	$-0.428744\pi$
$127$	5.00348	0.443987	0.221993	+	0.975048i	$0.428744\pi$
$128$	−8.72872	−0.771517
$129$	0	0
$130$	−0.687252	−0.0602760
$131$	19.0550	1.66485	0.832423	−	0.554140i	$-0.186953\pi$
$131$	19.0550	1.66485	0.832423	+	0.554140i	$0.186953\pi$
$132$	0	0
$133$	−0.532481	−0.0461720
$134$	−6.11422	−0.528188
$135$	0	0
$136$	−1.01607	−0.0871277
$137$	−18.4521	−1.57647	−0.788237	−	0.615372i	$-0.789006\pi$
$137$	−18.4521	−1.57647	−0.788237	+	0.615372i	$0.789006\pi$
$138$	0	0
$139$	14.6161	1.23972	0.619861	−	0.784711i	$-0.287189\pi$
$139$	14.6161	1.23972	0.619861	+	0.784711i	$0.287189\pi$
$140$	−0.292313	−0.0247050
$141$	0	0
$142$	7.43882	0.624252
$143$	−0.801491	−0.0670241
$144$	0	0
$145$	0.966063	0.0802271
$146$	6.03345	0.499332
$147$	0	0
$148$	2.90389	0.238698
$149$	4.26078	0.349057	0.174528	−	0.984652i	$-0.444160\pi$
$149$	4.26078	0.349057	0.174528	+	0.984652i	$0.444160\pi$
$150$	0	0
$151$	19.9980	1.62741	0.813707	−	0.581276i	$-0.197446\pi$
$151$	19.9980	1.62741	0.813707	+	0.581276i	$0.197446\pi$
$152$	−7.07726	−0.574041
$153$	0	0
$154$	0.163465	0.0131724
$155$	−2.21338	−0.177783
$156$	0	0
$157$	0.492192	0.0392812	0.0196406	−	0.999807i	$-0.493748\pi$
$157$	0.492192	0.0392812	0.0196406	+	0.999807i	$0.493748\pi$
$158$	5.97150	0.475067
$159$	0	0
$160$	−6.20342	−0.490423
$161$	0.595095	0.0469000
$162$	0	0
$163$	−1.93731	−0.151742	−0.0758708	−	0.997118i	$-0.524174\pi$
$163$	−1.93731	−0.151742	−0.0758708	+	0.997118i	$0.524174\pi$
$164$	−2.47753	−0.193463
$165$	0	0
$166$	6.88476	0.534361
$167$	−14.8338	−1.14788	−0.573938	−	0.818899i	$-0.694585\pi$
$167$	−14.8338	−1.14788	−0.573938	+	0.818899i	$0.694585\pi$
$168$	0	0
$169$	−12.3576	−0.950586
$170$	−0.322857	−0.0247620
$171$	0	0
$172$	3.04529	0.232201
$173$	10.3859	0.789623	0.394812	−	0.918762i	$-0.370810\pi$
$173$	10.3859	0.789623	0.394812	+	0.918762i	$0.370810\pi$
$174$	0	0
$175$	0.784874	0.0593309
$176$	−0.530979	−0.0400240
$177$	0	0
$178$	2.39800	0.179738
$179$	−16.9049	−1.26353	−0.631765	−	0.775160i	$-0.717669\pi$
$179$	−16.9049	−1.26353	−0.631765	+	0.775160i	$0.717669\pi$
$180$	0	0
$181$	2.91712	0.216828	0.108414	−	0.994106i	$-0.465423\pi$
$181$	2.91712	0.216828	0.108414	+	0.994106i	$0.465423\pi$
$182$	−0.131016	−0.00971154
$183$	0	0
$184$	7.90945	0.583093
$185$	2.28787	0.168207
$186$	0	0
$187$	−0.376525	−0.0275342
$188$	−0.926570	−0.0675771
$189$	0	0
$190$	−2.24880	−0.163145
$191$	−10.8546	−0.785410	−0.392705	−	0.919664i	$-0.628461\pi$
$191$	−10.8546	−0.785410	−0.392705	+	0.919664i	$0.628461\pi$
$192$	0	0
$193$	9.85294	0.709230	0.354615	−	0.935012i	$-0.384612\pi$
$193$	9.85294	0.709230	0.354615	+	0.935012i	$0.384612\pi$
$194$	14.3464	1.03001
$195$	0	0
$196$	9.40690	0.671921
$197$	−1.58403	−0.112857	−0.0564286	−	0.998407i	$-0.517971\pi$
$197$	−1.58403	−0.112857	−0.0564286	+	0.998407i	$0.517971\pi$
$198$	0	0
$199$	25.5545	1.81151	0.905756	−	0.423799i	$-0.139304\pi$
$199$	25.5545	1.81151	0.905756	+	0.423799i	$0.139304\pi$
$200$	10.4318	0.737641
$201$	0	0
$202$	3.63010	0.255413
$203$	0.184168	0.0129260
$204$	0	0
$205$	−1.95196	−0.136330
$206$	16.0067	1.11524
$207$	0	0
$208$	0.425575	0.0295083
$209$	−2.62261	−0.181409
$210$	0	0
$211$	16.1904	1.11460	0.557298	−	0.830313i	$-0.311838\pi$
$211$	16.1904	1.11460	0.557298	+	0.830313i	$0.311838\pi$
$212$	18.9866	1.30400
$213$	0	0
$214$	−5.30453	−0.362610
$215$	2.39927	0.163629
$216$	0	0
$217$	−0.421952	−0.0286440
$218$	15.4156	1.04408
$219$	0	0
$220$	−1.43972	−0.0970657
$221$	0.301781	0.0203000
$222$	0	0
$223$	−4.93169	−0.330250	−0.165125	−	0.986273i	$-0.552803\pi$
$223$	−4.93169	−0.330250	−0.165125	+	0.986273i	$0.552803\pi$
$224$	−1.18260	−0.0790159
$225$	0	0
$226$	2.28708	0.152134
$227$	2.59452	0.172204	0.0861022	−	0.996286i	$-0.472559\pi$
$227$	2.59452	0.172204	0.0861022	+	0.996286i	$0.472559\pi$
$228$	0	0
$229$	−6.75414	−0.446326	−0.223163	−	0.974781i	$-0.571638\pi$
$229$	−6.75414	−0.446326	−0.223163	+	0.974781i	$0.571638\pi$
$230$	2.51323	0.165717
$231$	0	0
$232$	2.44779	0.160705
$233$	27.8481	1.82439	0.912195	−	0.409757i	$-0.134386\pi$
$233$	27.8481	1.82439	0.912195	+	0.409757i	$0.134386\pi$
$234$	0	0
$235$	−0.730010	−0.0476206
$236$	10.4591	0.680830
$237$	0	0
$238$	−0.0615487	−0.00398961
$239$	−16.9202	−1.09448	−0.547240	−	0.836976i	$-0.684321\pi$
$239$	−16.9202	−1.09448	−0.547240	+	0.836976i	$0.684321\pi$
$240$	0	0
$241$	−1.06455	−0.0685739	−0.0342869	−	0.999412i	$-0.510916\pi$
$241$	−1.06455	−0.0685739	−0.0342869	+	0.999412i	$0.510916\pi$
$242$	0.805107	0.0517542
$243$	0	0
$244$	1.35180	0.0865403
$245$	7.41134	0.473493
$246$	0	0
$247$	2.10199	0.133747
$248$	−5.60821	−0.356121
$249$	0	0
$250$	7.60204	0.480795
$251$	3.19791	0.201850	0.100925	−	0.994894i	$-0.467820\pi$
$251$	3.19791	0.201850	0.100925	+	0.994894i	$0.467820\pi$
$252$	0	0
$253$	2.93099	0.184270
$254$	4.02833	0.252760
$255$	0	0
$256$	−14.2825	−0.892657
$257$	12.0296	0.750388	0.375194	−	0.926946i	$-0.377576\pi$
$257$	12.0296	0.750388	0.375194	+	0.926946i	$0.377576\pi$
$258$	0	0
$259$	0.436152	0.0271012
$260$	1.15392	0.0715631
$261$	0	0
$262$	15.3413	0.947791
$263$	3.98169	0.245521	0.122761	−	0.992436i	$-0.460825\pi$
$263$	3.98169	0.245521	0.122761	+	0.992436i	$0.460825\pi$
$264$	0	0
$265$	14.9588	0.918912
$266$	−0.428704	−0.0262855
$267$	0	0
$268$	10.2660	0.627095
$269$	−2.20633	−0.134522	−0.0672611	−	0.997735i	$-0.521426\pi$
$269$	−2.20633	−0.134522	−0.0672611	+	0.997735i	$0.521426\pi$
$270$	0	0
$271$	10.9268	0.663754	0.331877	−	0.943323i	$-0.392318\pi$
$271$	10.9268	0.663754	0.331877	+	0.943323i	$0.392318\pi$
$272$	0.199927	0.0121223
$273$	0	0
$274$	−14.8559	−0.897480
$275$	3.86570	0.233111
$276$	0	0
$277$	7.40923	0.445177	0.222589	−	0.974912i	$-0.428549\pi$
$277$	7.40923	0.445177	0.222589	+	0.974912i	$0.428549\pi$
$278$	11.7675	0.705770
$279$	0	0
$280$	−0.583535	−0.0348729
$281$	0.155459	0.00927391	0.00463696	−	0.999989i	$-0.498524\pi$
$281$	0.155459	0.00927391	0.00463696	+	0.999989i	$0.498524\pi$
$282$	0	0
$283$	−11.9494	−0.710316	−0.355158	−	0.934806i	$-0.615573\pi$
$283$	−11.9494	−0.710316	−0.355158	+	0.934806i	$0.615573\pi$
$284$	−12.4900	−0.741148
$285$	0	0
$286$	−0.645286	−0.0381565
$287$	−0.372115	−0.0219653
$288$	0	0
$289$	−16.8582	−0.991661
$290$	0.777784	0.0456730
$291$	0	0
$292$	−10.1304	−0.592836
$293$	−16.9701	−0.991404	−0.495702	−	0.868493i	$-0.665089\pi$
$293$	−16.9701	−0.991404	−0.495702	+	0.868493i	$0.665089\pi$
$294$	0	0
$295$	8.24034	0.479771
$296$	5.79694	0.336940
$297$	0	0
$298$	3.43038	0.198717
$299$	−2.34916	−0.135856
$300$	0	0
$301$	0.457390	0.0263635
$302$	16.1005	0.926480
$303$	0	0
$304$	1.39255	0.0798681
$305$	1.06503	0.0609837
$306$	0	0
$307$	13.3520	0.762041	0.381021	−	0.924567i	$-0.375573\pi$
$307$	13.3520	0.762041	0.381021	+	0.924567i	$0.375573\pi$
$308$	−0.274464	−0.0156390
$309$	0	0
$310$	−1.78201	−0.101211
$311$	12.6506	0.717352	0.358676	−	0.933462i	$-0.383228\pi$
$311$	12.6506	0.717352	0.358676	+	0.933462i	$0.383228\pi$
$312$	0	0
$313$	8.15405	0.460894	0.230447	−	0.973085i	$-0.425981\pi$
$313$	8.15405	0.460894	0.230447	+	0.973085i	$0.425981\pi$
$314$	0.396267	0.0223627
$315$	0	0
$316$	−10.0264	−0.564027
$317$	22.7512	1.27784	0.638918	−	0.769274i	$-0.279382\pi$
$317$	22.7512	1.27784	0.638918	+	0.769274i	$0.279382\pi$
$318$	0	0
$319$	0.907071	0.0507863
$320$	−3.86339	−0.215970
$321$	0	0
$322$	0.479115	0.0267000
$323$	0.987476	0.0549446
$324$	0	0
$325$	−3.09832	−0.171864
$326$	−1.55974	−0.0863860
$327$	0	0
$328$	−4.94582	−0.273087
$329$	−0.139167	−0.00767253
$330$	0	0
$331$	2.47607	0.136097	0.0680486	−	0.997682i	$-0.478323\pi$
$331$	2.47607	0.136097	0.0680486	+	0.997682i	$0.478323\pi$
$332$	−11.5598	−0.634424
$333$	0	0
$334$	−11.9428	−0.653481
$335$	8.08819	0.441905
$336$	0	0
$337$	−11.8667	−0.646419	−0.323210	−	0.946327i	$-0.604762\pi$
$337$	−11.8667	−0.646419	−0.323210	+	0.946327i	$0.604762\pi$
$338$	−9.94920	−0.541165
$339$	0	0
$340$	0.542089	0.0293989
$341$	−2.07822	−0.112542
$342$	0	0
$343$	2.83412	0.153028
$344$	6.07920	0.327769
$345$	0	0
$346$	8.36174	0.449530
$347$	−33.4479	−1.79558	−0.897789	−	0.440426i	$-0.854827\pi$
$347$	−33.4479	−1.79558	−0.897789	+	0.440426i	$0.854827\pi$
$348$	0	0
$349$	−19.7785	−1.05872	−0.529359	−	0.848398i	$-0.677567\pi$
$349$	−19.7785	−1.05872	−0.529359	+	0.848398i	$0.677567\pi$
$350$	0.631907	0.0337768
$351$	0	0
$352$	−5.82461	−0.310453
$353$	15.2833	0.813448	0.406724	−	0.913551i	$-0.366671\pi$
$353$	15.2833	0.813448	0.406724	+	0.913551i	$0.366671\pi$
$354$	0	0
$355$	−9.84043	−0.522276
$356$	−4.02633	−0.213395
$357$	0	0
$358$	−13.6102	−0.719324
$359$	−10.2459	−0.540759	−0.270380	−	0.962754i	$-0.587149\pi$
$359$	−10.2459	−0.540759	−0.270380	+	0.962754i	$0.587149\pi$
$360$	0	0
$361$	−12.1219	−0.637997
$362$	2.34859	0.123439
$363$	0	0
$364$	0.219980	0.0115301
$365$	−7.98135	−0.417763
$366$	0	0
$367$	11.5481	0.602807	0.301403	−	0.953497i	$-0.402545\pi$
$367$	11.5481	0.602807	0.301403	+	0.953497i	$0.402545\pi$
$368$	−1.55629	−0.0811274
$369$	0	0
$370$	1.84198	0.0957598
$371$	2.85171	0.148053
$372$	0	0
$373$	−26.2319	−1.35824	−0.679119	−	0.734028i	$-0.737638\pi$
$373$	−26.2319	−1.35824	−0.679119	+	0.734028i	$0.737638\pi$
$374$	−0.303143	−0.0156751
$375$	0	0
$376$	−1.84968	−0.0953900
$377$	−0.727010	−0.0374429
$378$	0	0
$379$	11.1404	0.572245	0.286123	−	0.958193i	$-0.407634\pi$
$379$	11.1404	0.572245	0.286123	+	0.958193i	$0.407634\pi$
$380$	3.77581	0.193695
$381$	0	0
$382$	−8.73910	−0.447131
$383$	−37.6696	−1.92483	−0.962413	−	0.271590i	$-0.912451\pi$
$383$	−37.6696	−1.92483	−0.962413	+	0.271590i	$0.912451\pi$
$384$	0	0
$385$	−0.216240	−0.0110206
$386$	7.93266	0.403762
$387$	0	0
$388$	−24.0882	−1.22289
$389$	10.9370	0.554527	0.277263	−	0.960794i	$-0.410573\pi$
$389$	10.9370	0.554527	0.277263	+	0.960794i	$0.410573\pi$
$390$	0	0
$391$	−1.10359	−0.0558110
$392$	18.7787	0.948466
$393$	0	0
$394$	−1.27531	−0.0642492
$395$	−7.89939	−0.397461
$396$	0	0
$397$	33.2094	1.66673	0.833365	−	0.552722i	$-0.186411\pi$
$397$	33.2094	1.66673	0.833365	+	0.552722i	$0.186411\pi$
$398$	20.5741	1.03129
$399$	0	0
$400$	−2.05260	−0.102630
$401$	18.7562	0.936640	0.468320	−	0.883559i	$-0.344859\pi$
$401$	18.7562	0.936640	0.468320	+	0.883559i	$0.344859\pi$
$402$	0	0
$403$	1.66568	0.0829733
$404$	−6.09506	−0.303241
$405$	0	0
$406$	0.148274	0.00735874
$407$	2.14816	0.106480
$408$	0	0
$409$	5.97380	0.295385	0.147693	−	0.989033i	$-0.452815\pi$
$409$	5.97380	0.295385	0.147693	+	0.989033i	$0.452815\pi$
$410$	−1.57153	−0.0776125
$411$	0	0
$412$	−26.8758	−1.32408
$413$	1.57091	0.0772997
$414$	0	0
$415$	−9.10750	−0.447069
$416$	4.66838	0.228886
$417$	0	0
$418$	−2.11148	−0.103276
$419$	21.7376	1.06195	0.530976	−	0.847387i	$-0.321826\pi$
$419$	21.7376	1.06195	0.530976	+	0.847387i	$0.321826\pi$
$420$	0	0
$421$	−16.2177	−0.790402	−0.395201	−	0.918595i	$-0.629325\pi$
$421$	−16.2177	−0.790402	−0.395201	+	0.918595i	$0.629325\pi$
$422$	13.0350	0.634535
$423$	0	0
$424$	37.9023	1.84070
$425$	−1.45553	−0.0706037
$426$	0	0
$427$	0.203035	0.00982557
$428$	8.90650	0.430512
$429$	0	0
$430$	1.93167	0.0931532
$431$	31.0756	1.49686	0.748430	−	0.663214i	$-0.230808\pi$
$431$	31.0756	1.49686	0.748430	+	0.663214i	$0.230808\pi$
$432$	0	0
$433$	37.4627	1.80034	0.900171	−	0.435536i	$-0.143441\pi$
$433$	37.4627	1.80034	0.900171	+	0.435536i	$0.143441\pi$
$434$	−0.339717	−0.0163069
$435$	0	0
$436$	−25.8834	−1.23959
$437$	−7.68683	−0.367711
$438$	0	0
$439$	15.0607	0.718809	0.359405	−	0.933182i	$-0.382980\pi$
$439$	15.0607	0.718809	0.359405	+	0.933182i	$0.382980\pi$
$440$	−2.87406	−0.137015
$441$	0	0
$442$	0.242966	0.0115567
$443$	−7.50403	−0.356527	−0.178264	−	0.983983i	$-0.557048\pi$
$443$	−7.50403	−0.356527	−0.178264	+	0.983983i	$0.557048\pi$
$444$	0	0
$445$	−3.17220	−0.150376
$446$	−3.97054	−0.188010
$447$	0	0
$448$	−0.736506	−0.0347966
$449$	−13.5424	−0.639103	−0.319552	−	0.947569i	$-0.603532\pi$
$449$	−13.5424	−0.639103	−0.319552	+	0.947569i	$0.603532\pi$
$450$	0	0
$451$	−1.83276	−0.0863014
$452$	−3.84009	−0.180623
$453$	0	0
$454$	2.08887	0.0980353
$455$	0.173314	0.00812509
$456$	0	0
$457$	10.1840	0.476388	0.238194	−	0.971218i	$-0.423445\pi$
$457$	10.1840	0.476388	0.238194	+	0.971218i	$0.423445\pi$
$458$	−5.43780	−0.254092
$459$	0	0
$460$	−4.21980	−0.196749
$461$	14.0649	0.655067	0.327533	−	0.944840i	$-0.393783\pi$
$461$	14.0649	0.655067	0.327533	+	0.944840i	$0.393783\pi$
$462$	0	0
$463$	−27.3090	−1.26916	−0.634578	−	0.772859i	$-0.718826\pi$
$463$	−27.3090	−1.26916	−0.634578	+	0.772859i	$0.718826\pi$
$464$	−0.481636	−0.0223594
$465$	0	0
$466$	22.4207	1.03862
$467$	−16.1833	−0.748875	−0.374437	−	0.927252i	$-0.622164\pi$
$467$	−16.1833	−0.748875	−0.374437	+	0.927252i	$0.622164\pi$
$468$	0	0
$469$	1.54191	0.0711988
$470$	−0.587736	−0.0271102
$471$	0	0
$472$	20.8792	0.961041
$473$	2.25276	0.103582
$474$	0	0
$475$	−10.1382	−0.465173
$476$	0.103342	0.00473669
$477$	0	0
$478$	−13.6226	−0.623083
$479$	19.5798	0.894623	0.447311	−	0.894378i	$-0.352382\pi$
$479$	19.5798	0.894623	0.447311	+	0.894378i	$0.352382\pi$
$480$	0	0
$481$	−1.72173	−0.0785042
$482$	−0.857078	−0.0390389
$483$	0	0
$484$	−1.35180	−0.0614456
$485$	−18.9782	−0.861754
$486$	0	0
$487$	−25.6857	−1.16393	−0.581966	−	0.813213i	$-0.697716\pi$
$487$	−25.6857	−1.16393	−0.581966	+	0.813213i	$0.697716\pi$
$488$	2.69856	0.122158
$489$	0	0
$490$	5.96692	0.269558
$491$	−30.0246	−1.35499	−0.677497	−	0.735526i	$-0.736935\pi$
$491$	−30.0246	−1.35499	−0.677497	+	0.735526i	$0.736935\pi$
$492$	0	0
$493$	−0.341535	−0.0153820
$494$	1.69233	0.0761415
$495$	0	0
$496$	1.10349	0.0495482
$497$	−1.87595	−0.0841480
$498$	0	0
$499$	16.6668	0.746109	0.373054	−	0.927810i	$-0.378311\pi$
$499$	16.6668	0.746109	0.373054	+	0.927810i	$0.378311\pi$
$500$	−12.7641	−0.570828
$501$	0	0
$502$	2.57466	0.114912
$503$	27.2800	1.21635	0.608176	−	0.793802i	$-0.291901\pi$
$503$	27.2800	1.21635	0.608176	+	0.793802i	$0.291901\pi$
$504$	0	0
$505$	−4.80207	−0.213689
$506$	2.35976	0.104904
$507$	0	0
$508$	−6.76372	−0.300092
$509$	21.2805	0.943243	0.471622	−	0.881801i	$-0.343669\pi$
$509$	21.2805	0.943243	0.471622	+	0.881801i	$0.343669\pi$
$510$	0	0
$511$	−1.52154	−0.0673091
$512$	5.95850	0.263331
$513$	0	0
$514$	9.68514	0.427193
$515$	−21.1744	−0.933057
$516$	0	0
$517$	−0.685433	−0.0301453
$518$	0.351149	0.0154286
$519$	0	0
$520$	2.30353	0.101017
$521$	25.1670	1.10258	0.551292	−	0.834312i	$-0.314135\pi$
$521$	25.1670	1.10258	0.551292	+	0.834312i	$0.314135\pi$
$522$	0	0
$523$	20.0188	0.875362	0.437681	−	0.899130i	$-0.355800\pi$
$523$	20.0188	0.875362	0.437681	+	0.899130i	$0.355800\pi$
$524$	−25.7587	−1.12527
$525$	0	0
$526$	3.20568	0.139774
$527$	0.782502	0.0340863
$528$	0	0
$529$	−14.4093	−0.626491
$530$	12.0434	0.523134
$531$	0	0
$532$	0.719810	0.0312077
$533$	1.46894	0.0636270
$534$	0	0
$535$	7.01710	0.303376
$536$	20.4936	0.885191
$537$	0	0
$538$	−1.77633	−0.0765830
$539$	6.95878	0.299736
$540$	0	0
$541$	33.9981	1.46169	0.730847	−	0.682541i	$-0.239125\pi$
$541$	33.9981	1.46169	0.730847	+	0.682541i	$0.239125\pi$
$542$	8.79722	0.377873
$543$	0	0
$544$	2.19311	0.0940289
$545$	−20.3926	−0.873521
$546$	0	0
$547$	−17.6728	−0.755636	−0.377818	−	0.925880i	$-0.623325\pi$
$547$	−17.6728	−0.755636	−0.377818	+	0.925880i	$0.623325\pi$
$548$	24.9437	1.06554
$549$	0	0
$550$	3.11230	0.132709
$551$	−2.37889	−0.101344
$552$	0	0
$553$	−1.50592	−0.0640381
$554$	5.96522	0.253438
$555$	0	0
$556$	−19.7581	−0.837931
$557$	−40.6753	−1.72347	−0.861734	−	0.507360i	$-0.830621\pi$
$557$	−40.6753	−1.72347	−0.861734	+	0.507360i	$0.830621\pi$
$558$	0	0
$559$	−1.80557	−0.0763673
$560$	0.114819	0.00485197
$561$	0	0
$562$	0.125161	0.00527961
$563$	−43.3214	−1.82578	−0.912890	−	0.408207i	$-0.866154\pi$
$563$	−43.3214	−1.82578	−0.912890	+	0.408207i	$0.866154\pi$
$564$	0	0
$565$	−3.02546	−0.127282
$566$	−9.62051	−0.404380
$567$	0	0
$568$	−24.9335	−1.04618
$569$	26.7783	1.12260	0.561302	−	0.827611i	$-0.310301\pi$
$569$	26.7783	1.12260	0.561302	+	0.827611i	$0.310301\pi$
$570$	0	0
$571$	29.9130	1.25182	0.625910	−	0.779896i	$-0.284728\pi$
$571$	29.9130	1.25182	0.625910	+	0.779896i	$0.284728\pi$
$572$	1.08346	0.0453017
$573$	0	0
$574$	−0.299593	−0.0125048
$575$	11.3303	0.472508
$576$	0	0
$577$	−19.3583	−0.805897	−0.402948	−	0.915223i	$-0.632015\pi$
$577$	−19.3583	−0.805897	−0.402948	+	0.915223i	$0.632015\pi$
$578$	−13.5727	−0.564549
$579$	0	0
$580$	−1.30593	−0.0542257
$581$	−1.73623	−0.0720309
$582$	0	0
$583$	14.0454	0.581700
$584$	−20.2230	−0.836832
$585$	0	0
$586$	−13.6627	−0.564402
$587$	33.1146	1.36679	0.683394	−	0.730050i	$-0.260503\pi$
$587$	33.1146	1.36679	0.683394	+	0.730050i	$0.260503\pi$
$588$	0	0
$589$	5.45036	0.224578
$590$	6.63435	0.273132
$591$	0	0
$592$	−1.14063	−0.0468795
$593$	11.3857	0.467555	0.233778	−	0.972290i	$-0.424891\pi$
$593$	11.3857	0.467555	0.233778	+	0.972290i	$0.424891\pi$
$594$	0	0
$595$	0.0814196	0.00333788
$596$	−5.75974	−0.235928
$597$	0	0
$598$	−1.89133	−0.0773421
$599$	−14.0221	−0.572927	−0.286463	−	0.958091i	$-0.592480\pi$
$599$	−14.0221	−0.572927	−0.286463	+	0.958091i	$0.592480\pi$
$600$	0	0
$601$	1.14648	0.0467660	0.0233830	−	0.999727i	$-0.492556\pi$
$601$	1.14648	0.0467660	0.0233830	+	0.999727i	$0.492556\pi$
$602$	0.368247	0.0150086
$603$	0	0
$604$	−27.0333	−1.09997
$605$	−1.06503	−0.0432998
$606$	0	0
$607$	−15.6771	−0.636315	−0.318157	−	0.948038i	$-0.603064\pi$
$607$	−15.6771	−0.636315	−0.318157	+	0.948038i	$0.603064\pi$
$608$	15.2757	0.619510
$609$	0	0
$610$	0.857467	0.0347178
$611$	0.549368	0.0222251
$612$	0	0
$613$	18.6826	0.754584	0.377292	−	0.926094i	$-0.376855\pi$
$613$	18.6826	0.754584	0.377292	+	0.926094i	$0.376855\pi$
$614$	10.7498	0.433827
$615$	0	0
$616$	−0.547903	−0.0220756
$617$	−3.28234	−0.132142	−0.0660709	−	0.997815i	$-0.521046\pi$
$617$	−3.28234	−0.132142	−0.0660709	+	0.997815i	$0.521046\pi$
$618$	0	0
$619$	−20.4789	−0.823116	−0.411558	−	0.911383i	$-0.635015\pi$
$619$	−20.4789	−0.823116	−0.411558	+	0.911383i	$0.635015\pi$
$620$	2.99205	0.120164
$621$	0	0
$622$	10.1851	0.408386
$623$	−0.604738	−0.0242283
$624$	0	0
$625$	9.27215	0.370886
$626$	6.56488	0.262385
$627$	0	0
$628$	−0.665347	−0.0265502
$629$	−0.808836	−0.0322504
$630$	0	0
$631$	−21.1120	−0.840454	−0.420227	−	0.907419i	$-0.638050\pi$
$631$	−21.1120	−0.840454	−0.420227	+	0.907419i	$0.638050\pi$
$632$	−20.0153	−0.796165
$633$	0	0
$634$	18.3172	0.727468
$635$	−5.32888	−0.211470
$636$	0	0
$637$	−5.57740	−0.220985
$638$	0.730289	0.0289124
$639$	0	0
$640$	9.29639	0.367472
$641$	28.2912	1.11743	0.558717	−	0.829358i	$-0.311294\pi$
$641$	28.2912	1.11743	0.558717	+	0.829358i	$0.311294\pi$
$642$	0	0
$643$	−20.9767	−0.827238	−0.413619	−	0.910450i	$-0.635736\pi$
$643$	−20.9767	−0.827238	−0.413619	+	0.910450i	$0.635736\pi$
$644$	−0.804451	−0.0316998
$645$	0	0
$646$	0.795024	0.0312798
$647$	−20.5640	−0.808456	−0.404228	−	0.914658i	$-0.632460\pi$
$647$	−20.5640	−0.808456	−0.404228	+	0.914658i	$0.632460\pi$
$648$	0	0
$649$	7.73715	0.303710
$650$	−2.49448	−0.0978416
$651$	0	0
$652$	2.61886	0.102562
$653$	−3.95699	−0.154849	−0.0774246	−	0.996998i	$-0.524670\pi$
$653$	−3.95699	−0.154849	−0.0774246	+	0.996998i	$0.524670\pi$
$654$	0	0
$655$	−20.2943	−0.792963
$656$	0.973158	0.0379954
$657$	0	0
$658$	−0.112044	−0.00436794
$659$	13.5419	0.527519	0.263759	−	0.964589i	$-0.415038\pi$
$659$	13.5419	0.527519	0.263759	+	0.964589i	$0.415038\pi$
$660$	0	0
$661$	−37.6849	−1.46577	−0.732886	−	0.680351i	$-0.761827\pi$
$661$	−37.6849	−1.46577	−0.732886	+	0.680351i	$0.761827\pi$
$662$	1.99350	0.0774797
$663$	0	0
$664$	−23.0764	−0.895536
$665$	0.567111	0.0219916
$666$	0	0
$667$	2.65862	0.102942
$668$	20.0524	0.775851
$669$	0	0
$670$	6.51185	0.251575
$671$	1.00000	0.0386046
$672$	0	0
$673$	46.2878	1.78427	0.892133	−	0.451774i	$-0.149209\pi$
$673$	46.2878	1.78427	0.892133	+	0.451774i	$0.149209\pi$
$674$	−9.55394	−0.368004
$675$	0	0
$676$	16.7051	0.642502
$677$	−26.6561	−1.02448	−0.512239	−	0.858843i	$-0.671184\pi$
$677$	−26.6561	−1.02448	−0.512239	+	0.858843i	$0.671184\pi$
$678$	0	0
$679$	−3.61795	−0.138844
$680$	1.08215	0.0414987
$681$	0	0
$682$	−1.67319	−0.0640698
$683$	−27.7294	−1.06104	−0.530519	−	0.847673i	$-0.678003\pi$
$683$	−27.7294	−1.06104	−0.530519	+	0.847673i	$0.678003\pi$
$684$	0	0
$685$	19.6522	0.750871
$686$	2.28177	0.0871185
$687$	0	0
$688$	−1.19617	−0.0456034
$689$	−11.2572	−0.428867
$690$	0	0
$691$	−14.4148	−0.548365	−0.274183	−	0.961678i	$-0.588407\pi$
$691$	−14.4148	−0.548365	−0.274183	+	0.961678i	$0.588407\pi$
$692$	−14.0397	−0.533708
$693$	0	0
$694$	−26.9291	−1.02222
$695$	−15.5667	−0.590477
$696$	0	0
$697$	0.690080	0.0261387
$698$	−15.9238	−0.602724
$699$	0	0
$700$	−1.06099	−0.0401018
$701$	21.9567	0.829295	0.414647	−	0.909982i	$-0.363905\pi$
$701$	21.9567	0.829295	0.414647	+	0.909982i	$0.363905\pi$
$702$	0	0
$703$	−5.63378	−0.212482
$704$	−3.62748	−0.136716
$705$	0	0
$706$	12.3047	0.463093
$707$	−0.915453	−0.0344292
$708$	0	0
$709$	−5.41410	−0.203331	−0.101665	−	0.994819i	$-0.532417\pi$
$709$	−5.41410	−0.203331	−0.101665	+	0.994819i	$0.532417\pi$
$710$	−7.92260	−0.297330
$711$	0	0
$712$	−8.03763	−0.301223
$713$	−6.09125	−0.228119
$714$	0	0
$715$	0.853616	0.0319234
$716$	22.8521	0.854023
$717$	0	0
$718$	−8.24906	−0.307852
$719$	−34.7613	−1.29638	−0.648188	−	0.761480i	$-0.724473\pi$
$719$	−34.7613	−1.29638	−0.648188	+	0.761480i	$0.724473\pi$
$720$	0	0
$721$	−4.03664	−0.150332
$722$	−9.75946	−0.363209
$723$	0	0
$724$	−3.94337	−0.146554
$725$	3.50647	0.130227
$726$	0	0
$727$	34.1852	1.26786	0.633930	−	0.773390i	$-0.281441\pi$
$727$	34.1852	1.26786	0.633930	+	0.773390i	$0.281441\pi$
$728$	0.439139	0.0162756
$729$	0	0
$730$	−6.42584	−0.237831
$731$	−0.848220	−0.0313725
$732$	0	0
$733$	−21.4554	−0.792474	−0.396237	−	0.918148i	$-0.629684\pi$
$733$	−21.4554	−0.792474	−0.396237	+	0.918148i	$0.629684\pi$
$734$	9.29746	0.343176
$735$	0	0
$736$	−17.0719	−0.629278
$737$	7.59429	0.279739
$738$	0	0
$739$	33.6003	1.23601	0.618004	−	0.786175i	$-0.287942\pi$
$739$	33.6003	1.23601	0.618004	+	0.786175i	$0.287942\pi$
$740$	−3.09274	−0.113692
$741$	0	0
$742$	2.29593	0.0842862
$743$	−3.15148	−0.115617	−0.0578083	−	0.998328i	$-0.518411\pi$
$743$	−3.15148	−0.115617	−0.0578083	+	0.998328i	$0.518411\pi$
$744$	0	0
$745$	−4.53788	−0.166255
$746$	−21.1195	−0.773240
$747$	0	0
$748$	0.508987	0.0186104
$749$	1.33772	0.0488792
$750$	0	0
$751$	−21.1165	−0.770551	−0.385276	−	0.922802i	$-0.625894\pi$
$751$	−21.1165	−0.770551	−0.385276	+	0.922802i	$0.625894\pi$
$752$	0.363950	0.0132719
$753$	0	0
$754$	−0.585320	−0.0213161
$755$	−21.2986	−0.775134
$756$	0	0
$757$	25.3065	0.919779	0.459890	−	0.887976i	$-0.347889\pi$
$757$	25.3065	0.919779	0.459890	+	0.887976i	$0.347889\pi$
$758$	8.96923	0.325777
$759$	0	0
$760$	7.53752	0.273415
$761$	26.3449	0.955000	0.477500	−	0.878632i	$-0.341543\pi$
$761$	26.3449	0.955000	0.477500	+	0.878632i	$0.341543\pi$
$762$	0	0
$763$	−3.88758	−0.140740
$764$	14.6733	0.530860
$765$	0	0
$766$	−30.3280	−1.09580
$767$	−6.20126	−0.223914
$768$	0	0
$769$	−0.854669	−0.0308202	−0.0154101	−	0.999881i	$-0.504905\pi$
$769$	−0.854669	−0.0308202	−0.0154101	+	0.999881i	$0.504905\pi$
$770$	−0.174096	−0.00627398
$771$	0	0
$772$	−13.3192	−0.479370
$773$	−14.3339	−0.515555	−0.257778	−	0.966204i	$-0.582990\pi$
$773$	−14.3339	−0.515555	−0.257778	+	0.966204i	$0.582990\pi$
$774$	0	0
$775$	−8.03378	−0.288582
$776$	−48.0864	−1.72620
$777$	0	0
$778$	8.80543	0.315690
$779$	4.80661	0.172215
$780$	0	0
$781$	−9.23954	−0.330617
$782$	−0.888508	−0.0317730
$783$	0	0
$784$	−3.69496	−0.131963
$785$	−0.524202	−0.0187096
$786$	0	0
$787$	46.4807	1.65686	0.828430	−	0.560093i	$-0.189235\pi$
$787$	46.4807	1.65686	0.828430	+	0.560093i	$0.189235\pi$
$788$	2.14129	0.0762804
$789$	0	0
$790$	−6.35985	−0.226273
$791$	−0.576766	−0.0205074
$792$	0	0
$793$	−0.801491	−0.0284618
$794$	26.7371	0.948864
$795$	0	0
$796$	−34.5447	−1.22440
$797$	13.5176	0.478817	0.239409	−	0.970919i	$-0.423046\pi$
$797$	13.5176	0.478817	0.239409	+	0.970919i	$0.423046\pi$
$798$	0	0
$799$	0.258082	0.00913030
$800$	−22.5162	−0.796068
$801$	0	0
$802$	15.1007	0.533226
$803$	−7.49398	−0.264457
$804$	0	0
$805$	−0.633797	−0.0223384
$806$	1.34105	0.0472364
$807$	0	0
$808$	−12.1674	−0.428046
$809$	−44.3814	−1.56037	−0.780183	−	0.625551i	$-0.784874\pi$
$809$	−44.3814	−1.56037	−0.780183	+	0.625551i	$0.784874\pi$
$810$	0	0
$811$	−2.02464	−0.0710946	−0.0355473	−	0.999368i	$-0.511317\pi$
$811$	−2.02464	−0.0710946	−0.0355473	+	0.999368i	$0.511317\pi$
$812$	−0.248958	−0.00873672
$813$	0	0
$814$	1.72950	0.0606189
$815$	2.06330	0.0722743
$816$	0	0
$817$	−5.90810	−0.206698
$818$	4.80954	0.168162
$819$	0	0
$820$	2.63866	0.0921460
$821$	−26.4603	−0.923472	−0.461736	−	0.887017i	$-0.652773\pi$
$821$	−26.4603	−0.923472	−0.461736	+	0.887017i	$0.652773\pi$
$822$	0	0
$823$	−40.6265	−1.41615	−0.708075	−	0.706137i	$-0.750436\pi$
$823$	−40.6265	−1.41615	−0.708075	+	0.706137i	$0.750436\pi$
$824$	−53.6513	−1.86903
$825$	0	0
$826$	1.26475	0.0440064
$827$	39.7573	1.38250	0.691248	−	0.722617i	$-0.257061\pi$
$827$	39.7573	1.38250	0.691248	+	0.722617i	$0.257061\pi$
$828$	0	0
$829$	−8.76395	−0.304385	−0.152192	−	0.988351i	$-0.548633\pi$
$829$	−8.76395	−0.304385	−0.152192	+	0.988351i	$0.548633\pi$
$830$	−7.33251	−0.254515
$831$	0	0
$832$	2.90739	0.100796
$833$	−2.62015	−0.0907829
$834$	0	0
$835$	15.7985	0.546731
$836$	3.54525	0.122615
$837$	0	0
$838$	17.5011	0.604565
$839$	0.750852	0.0259223	0.0129611	−	0.999916i	$-0.495874\pi$
$839$	0.750852	0.0259223	0.0129611	+	0.999916i	$0.495874\pi$
$840$	0	0
$841$	−28.1772	−0.971628
$842$	−13.0570	−0.449973
$843$	0	0
$844$	−21.8863	−0.753357
$845$	13.1613	0.452762
$846$	0	0
$847$	−0.203035	−0.00697637
$848$	−7.45779	−0.256102
$849$	0	0
$850$	−1.17186	−0.0401944
$851$	6.29624	0.215832
$852$	0	0
$853$	27.3127	0.935168	0.467584	−	0.883949i	$-0.345124\pi$
$853$	27.3127	0.935168	0.467584	+	0.883949i	$0.345124\pi$
$854$	0.163465	0.00559366
$855$	0	0
$856$	17.7798	0.607699
$857$	8.31434	0.284012	0.142006	−	0.989866i	$-0.454645\pi$
$857$	8.31434	0.284012	0.142006	+	0.989866i	$0.454645\pi$
$858$	0	0
$859$	27.0139	0.921703	0.460852	−	0.887477i	$-0.347544\pi$
$859$	27.0139	0.921703	0.460852	+	0.887477i	$0.347544\pi$
$860$	−3.24334	−0.110597
$861$	0	0
$862$	25.0192	0.852157
$863$	7.54619	0.256875	0.128438	−	0.991718i	$-0.459004\pi$
$863$	7.54619	0.256875	0.128438	+	0.991718i	$0.459004\pi$
$864$	0	0
$865$	−11.0613	−0.376096
$866$	30.1615	1.02493
$867$	0	0
$868$	0.570397	0.0193605
$869$	−7.41703	−0.251605
$870$	0	0
$871$	−6.08676	−0.206242
$872$	−51.6702	−1.74977
$873$	0	0
$874$	−6.18872	−0.209337
$875$	−1.91712	−0.0648104
$876$	0	0
$877$	−40.4243	−1.36503	−0.682515	−	0.730871i	$-0.739114\pi$
$877$	−40.4243	−1.36503	−0.682515	+	0.730871i	$0.739114\pi$
$878$	12.1255	0.409215
$879$	0	0
$880$	0.565511	0.0190634
$881$	24.8846	0.838383	0.419191	−	0.907898i	$-0.362314\pi$
$881$	24.8846	0.838383	0.419191	+	0.907898i	$0.362314\pi$
$882$	0	0
$883$	43.2827	1.45658	0.728289	−	0.685270i	$-0.240316\pi$
$883$	43.2827	1.45658	0.728289	+	0.685270i	$0.240316\pi$
$884$	−0.407949	−0.0137208
$885$	0	0
$886$	−6.04154	−0.202970
$887$	−10.2272	−0.343396	−0.171698	−	0.985150i	$-0.554925\pi$
$887$	−10.2272	−0.343396	−0.171698	+	0.985150i	$0.554925\pi$
$888$	0	0
$889$	−1.01588	−0.0340716
$890$	−2.55396	−0.0856088
$891$	0	0
$892$	6.66667	0.223217
$893$	1.79762	0.0601550
$894$	0	0
$895$	18.0043	0.601817
$896$	1.77224	0.0592063
$897$	0	0
$898$	−10.9030	−0.363839
$899$	−1.88510	−0.0628715
$900$	0	0
$901$	−5.28843	−0.176183
$902$	−1.47557	−0.0491311
$903$	0	0
$904$	−7.66584	−0.254962
$905$	−3.10683	−0.103275
$906$	0	0
$907$	−34.3774	−1.14148	−0.570741	−	0.821130i	$-0.693344\pi$
$907$	−34.3774	−1.14148	−0.570741	+	0.821130i	$0.693344\pi$
$908$	−3.50728	−0.116393
$909$	0	0
$910$	0.139536	0.00462559
$911$	20.2548	0.671072	0.335536	−	0.942027i	$-0.391083\pi$
$911$	20.2548	0.671072	0.335536	+	0.942027i	$0.391083\pi$
$912$	0	0
$913$	−8.55136	−0.283009
$914$	8.19921	0.271206
$915$	0	0
$916$	9.13027	0.301673
$917$	−3.86885	−0.127761
$918$	0	0
$919$	−26.9266	−0.888227	−0.444114	−	0.895970i	$-0.646481\pi$
$919$	−26.9266	−0.888227	−0.444114	+	0.895970i	$0.646481\pi$
$920$	−8.42384	−0.277726
$921$	0	0
$922$	11.3237	0.372927
$923$	7.40541	0.243752
$924$	0	0
$925$	8.30415	0.273039
$926$	−21.9866	−0.722526
$927$	0	0
$928$	−5.28334	−0.173434
$929$	−13.1495	−0.431421	−0.215711	−	0.976457i	$-0.569207\pi$
$929$	−13.1495	−0.431421	−0.215711	+	0.976457i	$0.569207\pi$
$930$	0	0
$931$	−18.2501	−0.598124
$932$	−37.6452	−1.23311
$933$	0	0
$934$	−13.0293	−0.426332
$935$	0.401012	0.0131145
$936$	0	0
$937$	19.9881	0.652981	0.326491	−	0.945200i	$-0.394134\pi$
$937$	19.9881	0.652981	0.326491	+	0.945200i	$0.394134\pi$
$938$	1.24140	0.0405332
$939$	0	0
$940$	0.986829	0.0321868
$941$	−24.3479	−0.793720	−0.396860	−	0.917879i	$-0.629900\pi$
$941$	−24.3479	−0.793720	−0.396860	+	0.917879i	$0.629900\pi$
$942$	0	0
$943$	−5.37181	−0.174930
$944$	−4.10826	−0.133713
$945$	0	0
$946$	1.81371	0.0589688
$947$	33.4192	1.08598	0.542989	−	0.839740i	$-0.317293\pi$
$947$	33.4192	1.08598	0.542989	+	0.839740i	$0.317293\pi$
$948$	0	0
$949$	6.00636	0.194975
$950$	−8.16234	−0.264821
$951$	0	0
$952$	0.206299	0.00668619
$953$	1.05530	0.0341847	0.0170923	−	0.999854i	$-0.494559\pi$
$953$	1.05530	0.0341847	0.0170923	+	0.999854i	$0.494559\pi$
$954$	0	0
$955$	11.5605	0.374089
$956$	22.8728	0.739761
$957$	0	0
$958$	15.7638	0.509305
$959$	3.74644	0.120979
$960$	0	0
$961$	−26.6810	−0.860677
$962$	−1.38618	−0.0446922
$963$	0	0
$964$	1.43907	0.0463492
$965$	−10.4937	−0.337805
$966$	0	0
$967$	15.5418	0.499791	0.249895	−	0.968273i	$-0.419604\pi$
$967$	15.5418	0.499791	0.249895	+	0.968273i	$0.419604\pi$
$968$	−2.69856	−0.0867350
$969$	0	0
$970$	−15.2795	−0.490594
$971$	27.9120	0.895738	0.447869	−	0.894099i	$-0.352183\pi$
$971$	27.9120	0.895738	0.447869	+	0.894099i	$0.352183\pi$
$972$	0	0
$973$	−2.96759	−0.0951365
$974$	−20.6797	−0.662622
$975$	0	0
$976$	−0.530979	−0.0169962
$977$	−10.2968	−0.329423	−0.164712	−	0.986342i	$-0.552669\pi$
$977$	−10.2968	−0.329423	−0.164712	+	0.986342i	$0.552669\pi$
$978$	0	0
$979$	−2.97849	−0.0951929
$980$	−10.0187	−0.320035
$981$	0	0
$982$	−24.1730	−0.771393
$983$	53.4728	1.70552	0.852759	−	0.522304i	$-0.174927\pi$
$983$	53.4728	1.70552	0.852759	+	0.522304i	$0.174927\pi$
$984$	0	0
$985$	1.68704	0.0537537
$986$	−0.274972	−0.00875689
$987$	0	0
$988$	−2.84148	−0.0903996
$989$	6.60282	0.209957
$990$	0	0
$991$	−22.7926	−0.724031	−0.362015	−	0.932172i	$-0.617911\pi$
$991$	−22.7926	−0.724031	−0.362015	+	0.932172i	$0.617911\pi$
$992$	12.1048	0.384329
$993$	0	0
$994$	−1.51034	−0.0479051
$995$	−27.2165	−0.862820
$996$	0	0
$997$	53.2933	1.68782	0.843908	−	0.536488i	$-0.180249\pi$
$997$	53.2933	1.68782	0.843908	+	0.536488i	$0.180249\pi$
$998$	13.4186	0.424757
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.i.1.8		13
3.2	odd	2		2013.2.a.e.1.6	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.e.1.6	✓	13	3.2	odd	2
6039.2.a.i.1.8		13	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)

\(f(q)\)	\(=\)	\(q+0.805107 q^{2} -1.35180 q^{4} -1.06503 q^{5} -0.203035 q^{7} -2.69856 q^{8} +O(q^{10})\)	\(q+0.805107 q^{2} -1.35180 q^{4} -1.06503 q^{5} -0.203035 q^{7} -2.69856 q^{8} -0.857467 q^{10} -1.00000 q^{11} +0.801491 q^{13} -0.163465 q^{14} +0.530979 q^{16} +0.376525 q^{17} +2.62261 q^{19} +1.43972 q^{20} -0.805107 q^{22} -2.93099 q^{23} -3.86570 q^{25} +0.645286 q^{26} +0.274464 q^{28} -0.907071 q^{29} +2.07822 q^{31} +5.82461 q^{32} +0.303143 q^{34} +0.216240 q^{35} -2.14816 q^{37} +2.11148 q^{38} +2.87406 q^{40} +1.83276 q^{41} -2.25276 q^{43} +1.35180 q^{44} -2.35976 q^{46} +0.685433 q^{47} -6.95878 q^{49} -3.11230 q^{50} -1.08346 q^{52} -14.0454 q^{53} +1.06503 q^{55} +0.547903 q^{56} -0.730289 q^{58} -7.73715 q^{59} -1.00000 q^{61} +1.67319 q^{62} +3.62748 q^{64} -0.853616 q^{65} -7.59429 q^{67} -0.508987 q^{68} +0.174096 q^{70} +9.23954 q^{71} +7.49398 q^{73} -1.72950 q^{74} -3.54525 q^{76} +0.203035 q^{77} +7.41703 q^{79} -0.565511 q^{80} +1.47557 q^{82} +8.55136 q^{83} -0.401012 q^{85} -1.81371 q^{86} +2.69856 q^{88} +2.97849 q^{89} -0.162731 q^{91} +3.96212 q^{92} +0.551846 q^{94} -2.79317 q^{95} +17.8193 q^{97} -5.60256 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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