LMFDB - Embedded newform 6039.2.a.n.1.16

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

Embedding invariants

Embedding label			1.16
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)

$f(q)$	$=$	$q+0.363962 q^{2} -1.86753 q^{4} -0.137887 q^{5} +3.90557 q^{7} -1.40764 q^{8} +O(q^{10})$	\(q+0.363962 q^{2} -1.86753 q^{4} -0.137887 q^{5} +3.90557 q^{7} -1.40764 q^{8} -0.0501855 q^{10} -1.00000 q^{11} -3.52513 q^{13} +1.42148 q^{14} +3.22274 q^{16} +7.69092 q^{17} -0.962815 q^{19} +0.257508 q^{20} -0.363962 q^{22} -2.43822 q^{23} -4.98099 q^{25} -1.28301 q^{26} -7.29378 q^{28} -8.63302 q^{29} -9.23184 q^{31} +3.98822 q^{32} +2.79920 q^{34} -0.538526 q^{35} +7.50603 q^{37} -0.350428 q^{38} +0.194094 q^{40} -3.88880 q^{41} +5.36784 q^{43} +1.86753 q^{44} -0.887418 q^{46} +4.27499 q^{47} +8.25349 q^{49} -1.81289 q^{50} +6.58329 q^{52} -6.82057 q^{53} +0.137887 q^{55} -5.49762 q^{56} -3.14209 q^{58} -11.3675 q^{59} -1.00000 q^{61} -3.36004 q^{62} -4.99391 q^{64} +0.486068 q^{65} +14.9404 q^{67} -14.3630 q^{68} -0.196003 q^{70} -3.34743 q^{71} +2.44036 q^{73} +2.73191 q^{74} +1.79809 q^{76} -3.90557 q^{77} -9.20719 q^{79} -0.444372 q^{80} -1.41538 q^{82} +3.69303 q^{83} -1.06047 q^{85} +1.95369 q^{86} +1.40764 q^{88} +15.9292 q^{89} -13.7676 q^{91} +4.55344 q^{92} +1.55593 q^{94} +0.132759 q^{95} -15.2937 q^{97} +3.00396 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8}+O(q^{10}) $ 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8	$ 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8} - 25 q^{11} + 4 q^{13} - 18 q^{14} + 21 q^{16} - 20 q^{17} + 14 q^{19} - 12 q^{20} + 5 q^{22} - 20 q^{23} + 13 q^{25} - 16 q^{26} - 14 q^{28} - 28 q^{29} - 12 q^{31} - 35 q^{32} + 6 q^{34} - 10 q^{35} - 8 q^{37} - 32 q^{38} + 24 q^{40} - 26 q^{41} + 18 q^{43} - 25 q^{44} + 4 q^{46} - 12 q^{47} + 23 q^{49} - 43 q^{50} + 22 q^{52} - 36 q^{53} + 4 q^{55} - 26 q^{56} - 20 q^{58} - 46 q^{59} - 25 q^{61} + 14 q^{62} - 13 q^{64} - 60 q^{65} - 20 q^{67} - 44 q^{68} - 20 q^{70} - 52 q^{71} + 6 q^{73} - 32 q^{74} - 4 q^{77} + 26 q^{79} - 52 q^{80} + 6 q^{82} - 38 q^{83} - 4 q^{85} - 34 q^{86} + 15 q^{88} - 82 q^{89} - 58 q^{91} - 36 q^{92} + 16 q^{94} - 30 q^{95} - 35 q^{98}+O(q^{100}) $ 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8 - 25 * q^11 + 4 * q^13 - 18 * q^14 + 21 * q^16 - 20 * q^17 + 14 * q^19 - 12 * q^20 + 5 * q^22 - 20 * q^23 + 13 * q^25 - 16 * q^26 - 14 * q^28 - 28 * q^29 - 12 * q^31 - 35 * q^32 + 6 * q^34 - 10 * q^35 - 8 * q^37 - 32 * q^38 + 24 * q^40 - 26 * q^41 + 18 * q^43 - 25 * q^44 + 4 * q^46 - 12 * q^47 + 23 * q^49 - 43 * q^50 + 22 * q^52 - 36 * q^53 + 4 * q^55 - 26 * q^56 - 20 * q^58 - 46 * q^59 - 25 * q^61 + 14 * q^62 - 13 * q^64 - 60 * q^65 - 20 * q^67 - 44 * q^68 - 20 * q^70 - 52 * q^71 + 6 * q^73 - 32 * q^74 - 4 * q^77 + 26 * q^79 - 52 * q^80 + 6 * q^82 - 38 * q^83 - 4 * q^85 - 34 * q^86 + 15 * q^88 - 82 * q^89 - 58 * q^91 - 36 * q^92 + 16 * q^94 - 30 * q^95 - 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 $

$2$	0.363962	0.257360	0.128680	−	0.991686i	$-0.458926\pi$
$2$	0.363962	0.257360	0.128680	+	0.991686i	$0.458926\pi$
$3$	0	0
$3$	0	0
$4$	−1.86753	−0.933766
$5$	−0.137887	−0.0616648	−0.0308324	−	0.999525i	$-0.509816\pi$
$5$	−0.137887	−0.0616648	−0.0308324	+	0.999525i	$0.509816\pi$
$6$	0	0
$7$	3.90557	1.47617	0.738084	−	0.674709i	$-0.235731\pi$
$7$	3.90557	1.47617	0.738084	+	0.674709i	$0.235731\pi$
$8$	−1.40764	−0.497674
$9$	0	0
$10$	−0.0501855	−0.0158700
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.52513	−0.977695	−0.488847	−	0.872369i	$-0.662583\pi$
$13$	−3.52513	−0.977695	−0.488847	+	0.872369i	$0.662583\pi$
$14$	1.42148	0.379907
$15$	0	0
$16$	3.22274	0.805684
$17$	7.69092	1.86532	0.932661	−	0.360754i	$-0.117481\pi$
$17$	7.69092	1.86532	0.932661	+	0.360754i	$0.117481\pi$
$18$	0	0
$19$	−0.962815	−0.220885	−0.110443	−	0.993883i	$-0.535227\pi$
$19$	−0.962815	−0.220885	−0.110443	+	0.993883i	$0.535227\pi$
$20$	0.257508	0.0575804
$21$	0	0
$22$	−0.363962	−0.0775970
$23$	−2.43822	−0.508403	−0.254202	−	0.967151i	$-0.581813\pi$
$23$	−2.43822	−0.508403	−0.254202	+	0.967151i	$0.581813\pi$
$24$	0	0
$25$	−4.98099	−0.996197
$26$	−1.28301	−0.251620
$27$	0	0
$28$	−7.29378	−1.37839
$29$	−8.63302	−1.60311	−0.801556	−	0.597920i	$-0.795994\pi$
$29$	−8.63302	−1.60311	−0.801556	+	0.597920i	$0.795994\pi$
$30$	0	0
$31$	−9.23184	−1.65809	−0.829043	−	0.559184i	$-0.811114\pi$
$31$	−9.23184	−1.65809	−0.829043	+	0.559184i	$0.811114\pi$
$32$	3.98822	0.705025
$33$	0	0
$34$	2.79920	0.480059
$35$	−0.538526	−0.0910275
$36$	0	0
$37$	7.50603	1.23398	0.616992	−	0.786969i	$-0.288351\pi$
$37$	7.50603	1.23398	0.616992	+	0.786969i	$0.288351\pi$
$38$	−0.350428	−0.0568470
$39$	0	0
$40$	0.194094	0.0306890
$41$	−3.88880	−0.607329	−0.303665	−	0.952779i	$-0.598210\pi$
$41$	−3.88880	−0.607329	−0.303665	+	0.952779i	$0.598210\pi$
$42$	0	0
$43$	5.36784	0.818588	0.409294	−	0.912403i	$-0.365775\pi$
$43$	5.36784	0.818588	0.409294	+	0.912403i	$0.365775\pi$
$44$	1.86753	0.281541
$45$	0	0
$46$	−0.887418	−0.130843
$47$	4.27499	0.623571	0.311786	−	0.950152i	$-0.399073\pi$
$47$	4.27499	0.623571	0.311786	+	0.950152i	$0.399073\pi$
$48$	0	0
$49$	8.25349	1.17907
$50$	−1.81289	−0.256381
$51$	0	0
$52$	6.58329	0.912938
$53$	−6.82057	−0.936877	−0.468439	−	0.883496i	$-0.655183\pi$
$53$	−6.82057	−0.936877	−0.468439	+	0.883496i	$0.655183\pi$
$54$	0	0
$55$	0.137887	0.0185926
$56$	−5.49762	−0.734650
$57$	0	0
$58$	−3.14209	−0.412577
$59$	−11.3675	−1.47992	−0.739960	−	0.672651i	$-0.765156\pi$
$59$	−11.3675	−1.47992	−0.739960	+	0.672651i	$0.765156\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−3.36004	−0.426725
$63$	0	0
$64$	−4.99391	−0.624239
$65$	0.486068	0.0602893
$66$	0	0
$67$	14.9404	1.82526	0.912631	−	0.408783i	$-0.134047\pi$
$67$	14.9404	1.82526	0.912631	+	0.408783i	$0.134047\pi$
$68$	−14.3630	−1.74177
$69$	0	0
$70$	−0.196003	−0.0234268
$71$	−3.34743	−0.397266	−0.198633	−	0.980074i	$-0.563650\pi$
$71$	−3.34743	−0.397266	−0.198633	+	0.980074i	$0.563650\pi$
$72$	0	0
$73$	2.44036	0.285623	0.142811	−	0.989750i	$-0.454386\pi$
$73$	2.44036	0.285623	0.142811	+	0.989750i	$0.454386\pi$
$74$	2.73191	0.317578
$75$	0	0
$76$	1.79809	0.206255
$77$	−3.90557	−0.445081
$78$	0	0
$79$	−9.20719	−1.03589	−0.517945	−	0.855414i	$-0.673303\pi$
$79$	−9.20719	−1.03589	−0.517945	+	0.855414i	$0.673303\pi$
$80$	−0.444372	−0.0496823
$81$	0	0
$82$	−1.41538	−0.156302
$83$	3.69303	0.405363	0.202681	−	0.979245i	$-0.435034\pi$
$83$	3.69303	0.405363	0.202681	+	0.979245i	$0.435034\pi$
$84$	0	0
$85$	−1.06047	−0.115025
$86$	1.95369	0.210672
$87$	0	0
$88$	1.40764	0.150054
$89$	15.9292	1.68850	0.844248	−	0.535953i	$-0.180047\pi$
$89$	15.9292	1.68850	0.844248	+	0.535953i	$0.180047\pi$
$90$	0	0
$91$	−13.7676	−1.44324
$92$	4.55344	0.474729
$93$	0	0
$94$	1.55593	0.160482
$95$	0.132759	0.0136208
$96$	0	0
$97$	−15.2937	−1.55284	−0.776422	−	0.630213i	$-0.782968\pi$
$97$	−15.2937	−1.55284	−0.776422	+	0.630213i	$0.782968\pi$
$98$	3.00396	0.303446
$99$	0	0
$100$	9.30215	0.930215
$101$	0.656362	0.0653104	0.0326552	−	0.999467i	$-0.489604\pi$
$101$	0.656362	0.0653104	0.0326552	+	0.999467i	$0.489604\pi$
$102$	0	0
$103$	−0.473670	−0.0466720	−0.0233360	−	0.999728i	$-0.507429\pi$
$103$	−0.473670	−0.0466720	−0.0233360	+	0.999728i	$0.507429\pi$
$104$	4.96209	0.486573
$105$	0	0
$106$	−2.48243	−0.241115
$107$	−4.65773	−0.450280	−0.225140	−	0.974326i	$-0.572284\pi$
$107$	−4.65773	−0.450280	−0.225140	+	0.974326i	$0.572284\pi$
$108$	0	0
$109$	6.68876	0.640667	0.320333	−	0.947305i	$-0.396205\pi$
$109$	6.68876	0.640667	0.320333	+	0.947305i	$0.396205\pi$
$110$	0.0501855	0.00478500
$111$	0	0
$112$	12.5866	1.18932
$113$	−12.2868	−1.15585	−0.577923	−	0.816092i	$-0.696136\pi$
$113$	−12.2868	−1.15585	−0.577923	+	0.816092i	$0.696136\pi$
$114$	0	0
$115$	0.336197	0.0313506
$116$	16.1224	1.49693
$117$	0	0
$118$	−4.13733	−0.380872
$119$	30.0374	2.75353
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.363962	−0.0329516
$123$	0	0
$124$	17.2407	1.54826
$125$	1.37624	0.123095
$126$	0	0
$127$	−8.52211	−0.756215	−0.378107	−	0.925762i	$-0.623425\pi$
$127$	−8.52211	−0.756215	−0.378107	+	0.925762i	$0.623425\pi$
$128$	−9.79404	−0.865679
$129$	0	0
$130$	0.176910	0.0155161
$131$	−12.3312	−1.07738	−0.538691	−	0.842504i	$-0.681081\pi$
$131$	−12.3312	−1.07738	−0.538691	+	0.842504i	$0.681081\pi$
$132$	0	0
$133$	−3.76034	−0.326063
$134$	5.43775	0.469750
$135$	0	0
$136$	−10.8260	−0.928322
$137$	−7.12602	−0.608817	−0.304408	−	0.952542i	$-0.598459\pi$
$137$	−7.12602	−0.608817	−0.304408	+	0.952542i	$0.598459\pi$
$138$	0	0
$139$	16.8105	1.42585	0.712926	−	0.701240i	$-0.247370\pi$
$139$	16.8105	1.42585	0.712926	+	0.701240i	$0.247370\pi$
$140$	1.00571	0.0849984
$141$	0	0
$142$	−1.21834	−0.102241
$143$	3.52513	0.294786
$144$	0	0
$145$	1.19038	0.0988555
$146$	0.888199	0.0735079
$147$	0	0
$148$	−14.0178	−1.15225
$149$	−4.38013	−0.358834	−0.179417	−	0.983773i	$-0.557421\pi$
$149$	−4.38013	−0.358834	−0.179417	+	0.983773i	$0.557421\pi$
$150$	0	0
$151$	11.2797	0.917929	0.458965	−	0.888454i	$-0.348220\pi$
$151$	11.2797	0.917929	0.458965	+	0.888454i	$0.348220\pi$
$152$	1.35529	0.109929
$153$	0	0
$154$	−1.42148	−0.114546
$155$	1.27295	0.102246
$156$	0	0
$157$	−6.66624	−0.532024	−0.266012	−	0.963970i	$-0.585706\pi$
$157$	−6.66624	−0.532024	−0.266012	+	0.963970i	$0.585706\pi$
$158$	−3.35107	−0.266597
$159$	0	0
$160$	−0.549923	−0.0434752
$161$	−9.52263	−0.750488
$162$	0	0
$163$	1.12423	0.0880565	0.0440283	−	0.999030i	$-0.485981\pi$
$163$	1.12423	0.0880565	0.0440283	+	0.999030i	$0.485981\pi$
$164$	7.26246	0.567103
$165$	0	0
$166$	1.34412	0.104324
$167$	−23.7766	−1.83989	−0.919945	−	0.392047i	$-0.871767\pi$
$167$	−23.7766	−1.83989	−0.919945	+	0.392047i	$0.871767\pi$
$168$	0	0
$169$	−0.573467	−0.0441129
$170$	−0.385973	−0.0296027
$171$	0	0
$172$	−10.0246	−0.764369
$173$	1.72906	0.131458	0.0657290	−	0.997838i	$-0.479063\pi$
$173$	1.72906	0.131458	0.0657290	+	0.997838i	$0.479063\pi$
$174$	0	0
$175$	−19.4536	−1.47055
$176$	−3.22274	−0.242923
$177$	0	0
$178$	5.79764	0.434552
$179$	9.58099	0.716117	0.358058	−	0.933699i	$-0.383439\pi$
$179$	9.58099	0.716117	0.358058	+	0.933699i	$0.383439\pi$
$180$	0	0
$181$	−7.33408	−0.545138	−0.272569	−	0.962136i	$-0.587873\pi$
$181$	−7.33408	−0.545138	−0.272569	+	0.962136i	$0.587873\pi$
$182$	−5.01090	−0.371433
$183$	0	0
$184$	3.43212	0.253019
$185$	−1.03498	−0.0760933
$186$	0	0
$187$	−7.69092	−0.562416
$188$	−7.98368	−0.582269
$189$	0	0
$190$	0.0483194	0.00350546
$191$	−13.9074	−1.00631	−0.503153	−	0.864197i	$-0.667827\pi$
$191$	−13.9074	−1.00631	−0.503153	+	0.864197i	$0.667827\pi$
$192$	0	0
$193$	−20.0547	−1.44357	−0.721784	−	0.692118i	$-0.756678\pi$
$193$	−20.0547	−1.44357	−0.721784	+	0.692118i	$0.756678\pi$
$194$	−5.56634	−0.399640
$195$	0	0
$196$	−15.4137	−1.10098
$197$	11.4932	0.818859	0.409430	−	0.912342i	$-0.365728\pi$
$197$	11.4932	0.818859	0.409430	+	0.912342i	$0.365728\pi$
$198$	0	0
$199$	8.64871	0.613091	0.306545	−	0.951856i	$-0.400827\pi$
$199$	8.64871	0.613091	0.306545	+	0.951856i	$0.400827\pi$
$200$	7.01141	0.495782
$201$	0	0
$202$	0.238891	0.0168083
$203$	−33.7169	−2.36646
$204$	0	0
$205$	0.536214	0.0374508
$206$	−0.172398	−0.0120115
$207$	0	0
$208$	−11.3606	−0.787713
$209$	0.962815	0.0665993
$210$	0	0
$211$	−4.03411	−0.277720	−0.138860	−	0.990312i	$-0.544344\pi$
$211$	−4.03411	−0.277720	−0.138860	+	0.990312i	$0.544344\pi$
$212$	12.7376	0.874824
$213$	0	0
$214$	−1.69524	−0.115884
$215$	−0.740153	−0.0504780
$216$	0	0
$217$	−36.0556	−2.44761
$218$	2.43445	0.164882
$219$	0	0
$220$	−0.257508	−0.0173612
$221$	−27.1115	−1.82372
$222$	0	0
$223$	−9.30345	−0.623005	−0.311503	−	0.950245i	$-0.600832\pi$
$223$	−9.30345	−0.623005	−0.311503	+	0.950245i	$0.600832\pi$
$224$	15.5763	1.04074
$225$	0	0
$226$	−4.47193	−0.297468
$227$	−19.6918	−1.30699	−0.653495	−	0.756931i	$-0.726698\pi$
$227$	−19.6918	−1.30699	−0.653495	+	0.756931i	$0.726698\pi$
$228$	0	0
$229$	14.1693	0.936331	0.468165	−	0.883641i	$-0.344915\pi$
$229$	14.1693	0.936331	0.468165	+	0.883641i	$0.344915\pi$
$230$	0.122363	0.00806838
$231$	0	0
$232$	12.1521	0.797827
$233$	−9.29608	−0.609006	−0.304503	−	0.952511i	$-0.598490\pi$
$233$	−9.29608	−0.609006	−0.304503	+	0.952511i	$0.598490\pi$
$234$	0	0
$235$	−0.589464	−0.0384524
$236$	21.2291	1.38190
$237$	0	0
$238$	10.9325	0.708648
$239$	−15.8148	−1.02298	−0.511488	−	0.859291i	$-0.670905\pi$
$239$	−15.8148	−1.02298	−0.511488	+	0.859291i	$0.670905\pi$
$240$	0	0
$241$	−17.9483	−1.15615	−0.578076	−	0.815983i	$-0.696196\pi$
$241$	−17.9483	−1.15615	−0.578076	+	0.815983i	$0.696196\pi$
$242$	0.363962	0.0233964
$243$	0	0
$244$	1.86753	0.119556
$245$	−1.13805	−0.0727071
$246$	0	0
$247$	3.39405	0.215958
$248$	12.9951	0.825187
$249$	0	0
$250$	0.500901	0.0316798
$251$	−24.1477	−1.52419	−0.762096	−	0.647464i	$-0.775830\pi$
$251$	−24.1477	−1.52419	−0.762096	+	0.647464i	$0.775830\pi$
$252$	0	0
$253$	2.43822	0.153289
$254$	−3.10172	−0.194620
$255$	0	0
$256$	6.42316	0.401448
$257$	−19.3994	−1.21010	−0.605050	−	0.796188i	$-0.706847\pi$
$257$	−19.3994	−1.21010	−0.605050	+	0.796188i	$0.706847\pi$
$258$	0	0
$259$	29.3154	1.82157
$260$	−0.907747	−0.0562961
$261$	0	0
$262$	−4.48809	−0.277275
$263$	−14.4099	−0.888552	−0.444276	−	0.895890i	$-0.646539\pi$
$263$	−14.4099	−0.888552	−0.444276	+	0.895890i	$0.646539\pi$
$264$	0	0
$265$	0.940465	0.0577723
$266$	−1.36862	−0.0839157
$267$	0	0
$268$	−27.9017	−1.70437
$269$	−12.0114	−0.732350	−0.366175	−	0.930546i	$-0.619333\pi$
$269$	−12.0114	−0.732350	−0.366175	+	0.930546i	$0.619333\pi$
$270$	0	0
$271$	25.3648	1.54080	0.770400	−	0.637561i	$-0.220056\pi$
$271$	25.3648	1.54080	0.770400	+	0.637561i	$0.220056\pi$
$272$	24.7858	1.50286
$273$	0	0
$274$	−2.59360	−0.156685
$275$	4.98099	0.300365
$276$	0	0
$277$	9.11127	0.547443	0.273721	−	0.961809i	$-0.411745\pi$
$277$	9.11127	0.547443	0.273721	+	0.961809i	$0.411745\pi$
$278$	6.11840	0.366957
$279$	0	0
$280$	0.758048	0.0453020
$281$	−23.5991	−1.40780	−0.703902	−	0.710297i	$-0.748561\pi$
$281$	−23.5991	−1.40780	−0.703902	+	0.710297i	$0.748561\pi$
$282$	0	0
$283$	−16.2155	−0.963914	−0.481957	−	0.876195i	$-0.660074\pi$
$283$	−16.2155	−0.963914	−0.481957	+	0.876195i	$0.660074\pi$
$284$	6.25142	0.370954
$285$	0	0
$286$	1.28301	0.0758662
$287$	−15.1880	−0.896519
$288$	0	0
$289$	42.1502	2.47942
$290$	0.433252	0.0254415
$291$	0	0
$292$	−4.55745	−0.266705
$293$	29.5108	1.72404	0.862020	−	0.506874i	$-0.169199\pi$
$293$	29.5108	1.72404	0.862020	+	0.506874i	$0.169199\pi$
$294$	0	0
$295$	1.56742	0.0912589
$296$	−10.5658	−0.614122
$297$	0	0
$298$	−1.59420	−0.0923496
$299$	8.59502	0.497063
$300$	0	0
$301$	20.9645	1.20837
$302$	4.10539	0.236238
$303$	0	0
$304$	−3.10290	−0.177964
$305$	0.137887	0.00789536
$306$	0	0
$307$	4.64032	0.264837	0.132419	−	0.991194i	$-0.457726\pi$
$307$	4.64032	0.264837	0.132419	+	0.991194i	$0.457726\pi$
$308$	7.29378	0.415602
$309$	0	0
$310$	0.463304	0.0263139
$311$	20.4315	1.15856	0.579281	−	0.815128i	$-0.303333\pi$
$311$	20.4315	1.15856	0.579281	+	0.815128i	$0.303333\pi$
$312$	0	0
$313$	10.8708	0.614455	0.307227	−	0.951636i	$-0.400599\pi$
$313$	10.8708	0.614455	0.307227	+	0.951636i	$0.400599\pi$
$314$	−2.42626	−0.136922
$315$	0	0
$316$	17.1947	0.967278
$317$	3.62111	0.203382	0.101691	−	0.994816i	$-0.467575\pi$
$317$	3.62111	0.203382	0.101691	+	0.994816i	$0.467575\pi$
$318$	0	0
$319$	8.63302	0.483356
$320$	0.688593	0.0384935
$321$	0	0
$322$	−3.46588	−0.193146
$323$	−7.40493	−0.412022
$324$	0	0
$325$	17.5586	0.973977
$326$	0.409177	0.0226622
$327$	0	0
$328$	5.47402	0.302252
$329$	16.6963	0.920495
$330$	0	0
$331$	17.8404	0.980594	0.490297	−	0.871555i	$-0.336888\pi$
$331$	17.8404	0.980594	0.490297	+	0.871555i	$0.336888\pi$
$332$	−6.89685	−0.378514
$333$	0	0
$334$	−8.65379	−0.473514
$335$	−2.06008	−0.112554
$336$	0	0
$337$	−13.9698	−0.760986	−0.380493	−	0.924784i	$-0.624246\pi$
$337$	−13.9698	−0.760986	−0.380493	+	0.924784i	$0.624246\pi$
$338$	−0.208720	−0.0113529
$339$	0	0
$340$	1.98047	0.107406
$341$	9.23184	0.499932
$342$	0	0
$343$	4.89559	0.264337
$344$	−7.55596	−0.407390
$345$	0	0
$346$	0.629313	0.0338321
$347$	−30.7978	−1.65331	−0.826657	−	0.562706i	$-0.809760\pi$
$347$	−30.7978	−1.65331	−0.826657	+	0.562706i	$0.809760\pi$
$348$	0	0
$349$	−14.1552	−0.757712	−0.378856	−	0.925456i	$-0.623682\pi$
$349$	−14.1552	−0.757712	−0.378856	+	0.925456i	$0.623682\pi$
$350$	−7.08037	−0.378462
$351$	0	0
$352$	−3.98822	−0.212573
$353$	17.8631	0.950758	0.475379	−	0.879781i	$-0.342311\pi$
$353$	17.8631	0.950758	0.475379	+	0.879781i	$0.342311\pi$
$354$	0	0
$355$	0.461565	0.0244973
$356$	−29.7484	−1.57666
$357$	0	0
$358$	3.48712	0.184300
$359$	9.79570	0.516998	0.258499	−	0.966012i	$-0.416772\pi$
$359$	9.79570	0.516998	0.258499	+	0.966012i	$0.416772\pi$
$360$	0	0
$361$	−18.0730	−0.951210
$362$	−2.66933	−0.140297
$363$	0	0
$364$	25.7115	1.34765
$365$	−0.336493	−0.0176129
$366$	0	0
$367$	−6.55836	−0.342344	−0.171172	−	0.985241i	$-0.554755\pi$
$367$	−6.55836	−0.342344	−0.171172	+	0.985241i	$0.554755\pi$
$368$	−7.85773	−0.409612
$369$	0	0
$370$	−0.376694	−0.0195834
$371$	−26.6382	−1.38299
$372$	0	0
$373$	−5.97621	−0.309436	−0.154718	−	0.987959i	$-0.549447\pi$
$373$	−5.97621	−0.309436	−0.154718	+	0.987959i	$0.549447\pi$
$374$	−2.79920	−0.144743
$375$	0	0
$376$	−6.01762	−0.310335
$377$	30.4325	1.56735
$378$	0	0
$379$	−0.362965	−0.0186443	−0.00932214	−	0.999957i	$-0.502967\pi$
$379$	−0.362965	−0.0186443	−0.00932214	+	0.999957i	$0.502967\pi$
$380$	−0.247932	−0.0127187
$381$	0	0
$382$	−5.06178	−0.258983
$383$	−14.5196	−0.741918	−0.370959	−	0.928649i	$-0.620971\pi$
$383$	−14.5196	−0.741918	−0.370959	+	0.928649i	$0.620971\pi$
$384$	0	0
$385$	0.538526	0.0274458
$386$	−7.29915	−0.371517
$387$	0	0
$388$	28.5615	1.44999
$389$	−3.84007	−0.194699	−0.0973495	−	0.995250i	$-0.531036\pi$
$389$	−3.84007	−0.194699	−0.0973495	+	0.995250i	$0.531036\pi$
$390$	0	0
$391$	−18.7521	−0.948335
$392$	−11.6179	−0.586793
$393$	0	0
$394$	4.18310	0.210742
$395$	1.26955	0.0638779
$396$	0	0
$397$	23.4382	1.17633	0.588164	−	0.808742i	$-0.299851\pi$
$397$	23.4382	1.17633	0.588164	+	0.808742i	$0.299851\pi$
$398$	3.14780	0.157785
$399$	0	0
$400$	−16.0524	−0.802621
$401$	−30.4718	−1.52169	−0.760844	−	0.648934i	$-0.775215\pi$
$401$	−30.4718	−1.52169	−0.760844	+	0.648934i	$0.775215\pi$
$402$	0	0
$403$	32.5434	1.62110
$404$	−1.22578	−0.0609846
$405$	0	0
$406$	−12.2717	−0.609033
$407$	−7.50603	−0.372060
$408$	0	0
$409$	−9.10164	−0.450047	−0.225024	−	0.974353i	$-0.572246\pi$
$409$	−9.10164	−0.450047	−0.225024	+	0.974353i	$0.572246\pi$
$410$	0.195162	0.00963834
$411$	0	0
$412$	0.884593	0.0435808
$413$	−44.3965	−2.18461
$414$	0	0
$415$	−0.509220	−0.0249966
$416$	−14.0590	−0.689299
$417$	0	0
$418$	0.350428	0.0171400
$419$	−25.2361	−1.23287	−0.616433	−	0.787408i	$-0.711423\pi$
$419$	−25.2361	−1.23287	−0.616433	+	0.787408i	$0.711423\pi$
$420$	0	0
$421$	33.6008	1.63760	0.818802	−	0.574076i	$-0.194639\pi$
$421$	33.6008	1.63760	0.818802	+	0.574076i	$0.194639\pi$
$422$	−1.46826	−0.0714739
$423$	0	0
$424$	9.60088	0.466260
$425$	−38.3084	−1.85823
$426$	0	0
$427$	−3.90557	−0.189004
$428$	8.69846	0.420456
$429$	0	0
$430$	−0.269388	−0.0129910
$431$	−16.1859	−0.779648	−0.389824	−	0.920889i	$-0.627464\pi$
$431$	−16.1859	−0.779648	−0.389824	+	0.920889i	$0.627464\pi$
$432$	0	0
$433$	−30.0102	−1.44220	−0.721100	−	0.692831i	$-0.756363\pi$
$433$	−30.0102	−1.44220	−0.721100	+	0.692831i	$0.756363\pi$
$434$	−13.1229	−0.629918
$435$	0	0
$436$	−12.4915	−0.598233
$437$	2.34755	0.112299
$438$	0	0
$439$	−4.59599	−0.219354	−0.109677	−	0.993967i	$-0.534982\pi$
$439$	−4.59599	−0.219354	−0.109677	+	0.993967i	$0.534982\pi$
$440$	−0.194094	−0.00925307
$441$	0	0
$442$	−9.86755	−0.469352
$443$	−23.1072	−1.09786	−0.548928	−	0.835869i	$-0.684964\pi$
$443$	−23.1072	−1.09786	−0.548928	+	0.835869i	$0.684964\pi$
$444$	0	0
$445$	−2.19643	−0.104121
$446$	−3.38610	−0.160337
$447$	0	0
$448$	−19.5041	−0.921481
$449$	−2.33237	−0.110071	−0.0550356	−	0.998484i	$-0.517527\pi$
$449$	−2.33237	−0.110071	−0.0550356	+	0.998484i	$0.517527\pi$
$450$	0	0
$451$	3.88880	0.183117
$452$	22.9460	1.07929
$453$	0	0
$454$	−7.16707	−0.336367
$455$	1.89837	0.0889971
$456$	0	0
$457$	12.3942	0.579778	0.289889	−	0.957060i	$-0.406382\pi$
$457$	12.3942	0.579778	0.289889	+	0.957060i	$0.406382\pi$
$458$	5.15707	0.240974
$459$	0	0
$460$	−0.627859	−0.0292741
$461$	−5.92712	−0.276054	−0.138027	−	0.990428i	$-0.544076\pi$
$461$	−5.92712	−0.276054	−0.138027	+	0.990428i	$0.544076\pi$
$462$	0	0
$463$	11.1549	0.518412	0.259206	−	0.965822i	$-0.416539\pi$
$463$	11.1549	0.518412	0.259206	+	0.965822i	$0.416539\pi$
$464$	−27.8220	−1.29160
$465$	0	0
$466$	−3.38342	−0.156734
$467$	21.1870	0.980418	0.490209	−	0.871605i	$-0.336920\pi$
$467$	21.1870	0.980418	0.490209	+	0.871605i	$0.336920\pi$
$468$	0	0
$469$	58.3509	2.69439
$470$	−0.214542	−0.00989610
$471$	0	0
$472$	16.0013	0.736518
$473$	−5.36784	−0.246813
$474$	0	0
$475$	4.79577	0.220045
$476$	−56.0959	−2.57115
$477$	0	0
$478$	−5.75599	−0.263273
$479$	29.3415	1.34065	0.670323	−	0.742069i	$-0.266155\pi$
$479$	29.3415	1.34065	0.670323	+	0.742069i	$0.266155\pi$
$480$	0	0
$481$	−26.4597	−1.20646
$482$	−6.53250	−0.297547
$483$	0	0
$484$	−1.86753	−0.0848878
$485$	2.10880	0.0957558
$486$	0	0
$487$	33.8198	1.53252	0.766261	−	0.642530i	$-0.222115\pi$
$487$	33.8198	1.53252	0.766261	+	0.642530i	$0.222115\pi$
$488$	1.40764	0.0637206
$489$	0	0
$490$	−0.414205	−0.0187119
$491$	31.4569	1.41963	0.709815	−	0.704388i	$-0.248779\pi$
$491$	31.4569	1.41963	0.709815	+	0.704388i	$0.248779\pi$
$492$	0	0
$493$	−66.3958	−2.99032
$494$	1.23531	0.0555790
$495$	0	0
$496$	−29.7518	−1.33589
$497$	−13.0736	−0.586432
$498$	0	0
$499$	−30.4237	−1.36195	−0.680977	−	0.732305i	$-0.738445\pi$
$499$	−30.4237	−1.36195	−0.680977	+	0.732305i	$0.738445\pi$
$500$	−2.57018	−0.114942
$501$	0	0
$502$	−8.78886	−0.392266
$503$	−34.5318	−1.53970	−0.769848	−	0.638227i	$-0.779668\pi$
$503$	−34.5318	−1.53970	−0.769848	+	0.638227i	$0.779668\pi$
$504$	0	0
$505$	−0.0905035	−0.00402735
$506$	0.887418	0.0394506
$507$	0	0
$508$	15.9153	0.706127
$509$	−24.7224	−1.09580	−0.547900	−	0.836544i	$-0.684573\pi$
$509$	−24.7224	−1.09580	−0.547900	+	0.836544i	$0.684573\pi$
$510$	0	0
$511$	9.53100	0.421627
$512$	21.9259	0.968996
$513$	0	0
$514$	−7.06064	−0.311431
$515$	0.0653127	0.00287802
$516$	0	0
$517$	−4.27499	−0.188014
$518$	10.6697	0.468799
$519$	0	0
$520$	−0.684206	−0.0300044
$521$	31.1697	1.36557	0.682785	−	0.730619i	$-0.260769\pi$
$521$	31.1697	1.36557	0.682785	+	0.730619i	$0.260769\pi$
$522$	0	0
$523$	−2.09188	−0.0914715	−0.0457357	−	0.998954i	$-0.514563\pi$
$523$	−2.09188	−0.0914715	−0.0457357	+	0.998954i	$0.514563\pi$
$524$	23.0289	1.00602
$525$	0	0
$526$	−5.24466	−0.228678
$527$	−71.0013	−3.09287
$528$	0	0
$529$	−17.0551	−0.741526
$530$	0.342294	0.0148683
$531$	0	0
$532$	7.02256	0.304467
$533$	13.7085	0.593783
$534$	0	0
$535$	0.642239	0.0277664
$536$	−21.0307	−0.908386
$537$	0	0
$538$	−4.37171	−0.188478
$539$	−8.25349	−0.355503
$540$	0	0
$541$	14.8404	0.638038	0.319019	−	0.947748i	$-0.396647\pi$
$541$	14.8404	0.638038	0.319019	+	0.947748i	$0.396647\pi$
$542$	9.23182	0.396541
$543$	0	0
$544$	30.6731	1.31510
$545$	−0.922290	−0.0395066
$546$	0	0
$547$	−10.6567	−0.455649	−0.227825	−	0.973702i	$-0.573161\pi$
$547$	−10.6567	−0.455649	−0.227825	+	0.973702i	$0.573161\pi$
$548$	13.3081	0.568492
$549$	0	0
$550$	1.81289	0.0773019
$551$	8.31200	0.354103
$552$	0	0
$553$	−35.9593	−1.52915
$554$	3.31616	0.140890
$555$	0	0
$556$	−31.3942	−1.33141
$557$	5.74103	0.243255	0.121628	−	0.992576i	$-0.461189\pi$
$557$	5.74103	0.243255	0.121628	+	0.992576i	$0.461189\pi$
$558$	0	0
$559$	−18.9223	−0.800329
$560$	−1.73553	−0.0733394
$561$	0	0
$562$	−8.58918	−0.362313
$563$	−5.32893	−0.224588	−0.112294	−	0.993675i	$-0.535820\pi$
$563$	−5.32893	−0.224588	−0.112294	+	0.993675i	$0.535820\pi$
$564$	0	0
$565$	1.69419	0.0712749
$566$	−5.90184	−0.248073
$567$	0	0
$568$	4.71195	0.197709
$569$	−4.02852	−0.168884	−0.0844422	−	0.996428i	$-0.526911\pi$
$569$	−4.02852	−0.168884	−0.0844422	+	0.996428i	$0.526911\pi$
$570$	0	0
$571$	13.0989	0.548173	0.274086	−	0.961705i	$-0.411625\pi$
$571$	13.0989	0.548173	0.274086	+	0.961705i	$0.411625\pi$
$572$	−6.58329	−0.275261
$573$	0	0
$574$	−5.52786	−0.230728
$575$	12.1447	0.506470
$576$	0	0
$577$	25.9281	1.07940	0.539700	−	0.841858i	$-0.318538\pi$
$577$	25.9281	1.07940	0.539700	+	0.841858i	$0.318538\pi$
$578$	15.3411	0.638105
$579$	0	0
$580$	−2.22307	−0.0923079
$581$	14.4234	0.598384
$582$	0	0
$583$	6.82057	0.282479
$584$	−3.43514	−0.142147
$585$	0	0
$586$	10.7408	0.443699
$587$	−28.2827	−1.16735	−0.583677	−	0.811986i	$-0.698387\pi$
$587$	−28.2827	−1.16735	−0.583677	+	0.811986i	$0.698387\pi$
$588$	0	0
$589$	8.88855	0.366247
$590$	0.570483	0.0234864
$591$	0	0
$592$	24.1900	0.994202
$593$	4.27991	0.175755	0.0878774	−	0.996131i	$-0.471992\pi$
$593$	4.27991	0.175755	0.0878774	+	0.996131i	$0.471992\pi$
$594$	0	0
$595$	−4.14176	−0.169796
$596$	8.18003	0.335067
$597$	0	0
$598$	3.12826	0.127924
$599$	−14.6271	−0.597648	−0.298824	−	0.954308i	$-0.596594\pi$
$599$	−14.6271	−0.597648	−0.298824	+	0.954308i	$0.596594\pi$
$600$	0	0
$601$	−46.8162	−1.90967	−0.954837	−	0.297130i	$-0.903970\pi$
$601$	−46.8162	−1.90967	−0.954837	+	0.297130i	$0.903970\pi$
$602$	7.63028	0.310987
$603$	0	0
$604$	−21.0652	−0.857131
$605$	−0.137887	−0.00560589
$606$	0	0
$607$	4.55698	0.184962	0.0924811	−	0.995714i	$-0.470520\pi$
$607$	4.55698	0.184962	0.0924811	+	0.995714i	$0.470520\pi$
$608$	−3.83992	−0.155729
$609$	0	0
$610$	0.0501855	0.00203195
$611$	−15.0699	−0.609662
$612$	0	0
$613$	9.18720	0.371068	0.185534	−	0.982638i	$-0.440599\pi$
$613$	9.18720	0.371068	0.185534	+	0.982638i	$0.440599\pi$
$614$	1.68890	0.0681585
$615$	0	0
$616$	5.49762	0.221505
$617$	27.7645	1.11776	0.558878	−	0.829250i	$-0.311232\pi$
$617$	27.7645	1.11776	0.558878	+	0.829250i	$0.311232\pi$
$618$	0	0
$619$	−0.329147	−0.0132295	−0.00661477	−	0.999978i	$-0.502106\pi$
$619$	−0.329147	−0.0132295	−0.00661477	+	0.999978i	$0.502106\pi$
$620$	−2.37727	−0.0954734
$621$	0	0
$622$	7.43628	0.298168
$623$	62.2128	2.49250
$624$	0	0
$625$	24.7152	0.988607
$626$	3.95657	0.158136
$627$	0	0
$628$	12.4494	0.496785
$629$	57.7283	2.30178
$630$	0	0
$631$	7.43789	0.296098	0.148049	−	0.988980i	$-0.452701\pi$
$631$	7.43789	0.296098	0.148049	+	0.988980i	$0.452701\pi$
$632$	12.9604	0.515536
$633$	0	0
$634$	1.31795	0.0523423
$635$	1.17508	0.0466318
$636$	0	0
$637$	−29.0946	−1.15277
$638$	3.14209	0.124397
$639$	0	0
$640$	1.35047	0.0533819
$641$	−4.64156	−0.183331	−0.0916653	−	0.995790i	$-0.529219\pi$
$641$	−4.64156	−0.183331	−0.0916653	+	0.995790i	$0.529219\pi$
$642$	0	0
$643$	3.15048	0.124243	0.0621214	−	0.998069i	$-0.480213\pi$
$643$	3.15048	0.124243	0.0621214	+	0.998069i	$0.480213\pi$
$644$	17.7838	0.700780
$645$	0	0
$646$	−2.69512	−0.106038
$647$	−22.3651	−0.879264	−0.439632	−	0.898178i	$-0.644891\pi$
$647$	−22.3651	−0.879264	−0.439632	+	0.898178i	$0.644891\pi$
$648$	0	0
$649$	11.3675	0.446213
$650$	6.39067	0.250663
$651$	0	0
$652$	−2.09954	−0.0822242
$653$	12.1413	0.475125	0.237562	−	0.971372i	$-0.423652\pi$
$653$	12.1413	0.475125	0.237562	+	0.971372i	$0.423652\pi$
$654$	0	0
$655$	1.70031	0.0664365
$656$	−12.5326	−0.489316
$657$	0	0
$658$	6.07681	0.236899
$659$	−9.60613	−0.374202	−0.187101	−	0.982341i	$-0.559909\pi$
$659$	−9.60613	−0.374202	−0.187101	+	0.982341i	$0.559909\pi$
$660$	0	0
$661$	33.4343	1.30044	0.650222	−	0.759744i	$-0.274676\pi$
$661$	33.4343	1.30044	0.650222	+	0.759744i	$0.274676\pi$
$662$	6.49321	0.252366
$663$	0	0
$664$	−5.19844	−0.201739
$665$	0.518501	0.0201066
$666$	0	0
$667$	21.0492	0.815027
$668$	44.4036	1.71803
$669$	0	0
$670$	−0.749793	−0.0289670
$671$	1.00000	0.0386046
$672$	0	0
$673$	3.11455	0.120057	0.0600285	−	0.998197i	$-0.480881\pi$
$673$	3.11455	0.120057	0.0600285	+	0.998197i	$0.480881\pi$
$674$	−5.08449	−0.195847
$675$	0	0
$676$	1.07097	0.0411911
$677$	−38.6509	−1.48547	−0.742737	−	0.669583i	$-0.766473\pi$
$677$	−38.6509	−1.48547	−0.742737	+	0.669583i	$0.766473\pi$
$678$	0	0
$679$	−59.7308	−2.29226
$680$	1.49276	0.0572448
$681$	0	0
$682$	3.36004	0.128663
$683$	47.7367	1.82659	0.913296	−	0.407296i	$-0.133528\pi$
$683$	47.7367	1.82659	0.913296	+	0.407296i	$0.133528\pi$
$684$	0	0
$685$	0.982582	0.0375425
$686$	1.78181	0.0680298
$687$	0	0
$688$	17.2991	0.659523
$689$	24.0434	0.915980
$690$	0	0
$691$	9.61034	0.365595	0.182797	−	0.983151i	$-0.441485\pi$
$691$	9.61034	0.365595	0.182797	+	0.983151i	$0.441485\pi$
$692$	−3.22908	−0.122751
$693$	0	0
$694$	−11.2092	−0.425497
$695$	−2.31795	−0.0879248
$696$	0	0
$697$	−29.9085	−1.13286
$698$	−5.15197	−0.195005
$699$	0	0
$700$	36.3302	1.37315
$701$	23.7197	0.895879	0.447940	−	0.894064i	$-0.352158\pi$
$701$	23.7197	0.895879	0.447940	+	0.894064i	$0.352158\pi$
$702$	0	0
$703$	−7.22693	−0.272569
$704$	4.99391	0.188215
$705$	0	0
$706$	6.50150	0.244687
$707$	2.56347	0.0964091
$708$	0	0
$709$	−31.1122	−1.16844	−0.584221	−	0.811595i	$-0.698600\pi$
$709$	−31.1122	−1.16844	−0.584221	+	0.811595i	$0.698600\pi$
$710$	0.167992	0.00630464
$711$	0	0
$712$	−22.4226	−0.840321
$713$	22.5092	0.842977
$714$	0	0
$715$	−0.486068	−0.0181779
$716$	−17.8928	−0.668685
$717$	0	0
$718$	3.56527	0.133055
$719$	35.7934	1.33487	0.667435	−	0.744668i	$-0.267392\pi$
$719$	35.7934	1.33487	0.667435	+	0.744668i	$0.267392\pi$
$720$	0	0
$721$	−1.84995	−0.0688957
$722$	−6.57788	−0.244803
$723$	0	0
$724$	13.6966	0.509031
$725$	43.0010	1.59702
$726$	0	0
$727$	30.5498	1.13303	0.566514	−	0.824052i	$-0.308292\pi$
$727$	30.5498	1.13303	0.566514	+	0.824052i	$0.308292\pi$
$728$	19.3798	0.718264
$729$	0	0
$730$	−0.122471	−0.00453285
$731$	41.2836	1.52693
$732$	0	0
$733$	−0.216017	−0.00797878	−0.00398939	−	0.999992i	$-0.501270\pi$
$733$	−0.216017	−0.00797878	−0.00398939	+	0.999992i	$0.501270\pi$
$734$	−2.38700	−0.0881056
$735$	0	0
$736$	−9.72415	−0.358437
$737$	−14.9404	−0.550337
$738$	0	0
$739$	−37.2594	−1.37061	−0.685305	−	0.728257i	$-0.740331\pi$
$739$	−37.2594	−1.37061	−0.685305	+	0.728257i	$0.740331\pi$
$740$	1.93286	0.0710534
$741$	0	0
$742$	−9.69531	−0.355926
$743$	−1.24583	−0.0457053	−0.0228526	−	0.999739i	$-0.507275\pi$
$743$	−1.24583	−0.0457053	−0.0228526	+	0.999739i	$0.507275\pi$
$744$	0	0
$745$	0.603961	0.0221274
$746$	−2.17511	−0.0796366
$747$	0	0
$748$	14.3630	0.525164
$749$	−18.1911	−0.664688
$750$	0	0
$751$	32.1595	1.17352	0.586758	−	0.809762i	$-0.300404\pi$
$751$	32.1595	1.17352	0.586758	+	0.809762i	$0.300404\pi$
$752$	13.7772	0.502401
$753$	0	0
$754$	11.0763	0.403374
$755$	−1.55532	−0.0566039
$756$	0	0
$757$	19.4244	0.705993	0.352997	−	0.935625i	$-0.385163\pi$
$757$	19.4244	0.705993	0.352997	+	0.935625i	$0.385163\pi$
$758$	−0.132106	−0.00479829
$759$	0	0
$760$	−0.186877	−0.00677873
$761$	−2.81443	−0.102023	−0.0510116	−	0.998698i	$-0.516245\pi$
$761$	−2.81443	−0.102023	−0.0510116	+	0.998698i	$0.516245\pi$
$762$	0	0
$763$	26.1234	0.945731
$764$	25.9726	0.939654
$765$	0	0
$766$	−5.28459	−0.190940
$767$	40.0718	1.44691
$768$	0	0
$769$	−51.2676	−1.84876	−0.924379	−	0.381476i	$-0.875416\pi$
$769$	−51.2676	−1.84876	−0.924379	+	0.381476i	$0.875416\pi$
$770$	0.196003	0.00706346
$771$	0	0
$772$	37.4528	1.34795
$773$	9.13158	0.328440	0.164220	−	0.986424i	$-0.447489\pi$
$773$	9.13158	0.328440	0.164220	+	0.986424i	$0.447489\pi$
$774$	0	0
$775$	45.9837	1.65178
$776$	21.5280	0.772811
$777$	0	0
$778$	−1.39764	−0.0501078
$779$	3.74420	0.134150
$780$	0	0
$781$	3.34743	0.119780
$782$	−6.82506	−0.244064
$783$	0	0
$784$	26.5988	0.949958
$785$	0.919185	0.0328071
$786$	0	0
$787$	28.6660	1.02183	0.510917	−	0.859630i	$-0.329306\pi$
$787$	28.6660	1.02183	0.510917	+	0.859630i	$0.329306\pi$
$788$	−21.4640	−0.764623
$789$	0	0
$790$	0.462067	0.0164396
$791$	−47.9870	−1.70622
$792$	0	0
$793$	3.52513	0.125181
$794$	8.53060	0.302740
$795$	0	0
$796$	−16.1517	−0.572483
$797$	14.0233	0.496731	0.248366	−	0.968666i	$-0.420107\pi$
$797$	14.0233	0.496731	0.248366	+	0.968666i	$0.420107\pi$
$798$	0	0
$799$	32.8786	1.16316
$800$	−19.8653	−0.702344
$801$	0	0
$802$	−11.0906	−0.391622
$803$	−2.44036	−0.0861185
$804$	0	0
$805$	1.31304	0.0462787
$806$	11.8446	0.417207
$807$	0	0
$808$	−0.923918	−0.0325033
$809$	−7.59101	−0.266886	−0.133443	−	0.991057i	$-0.542603\pi$
$809$	−7.59101	−0.266886	−0.133443	+	0.991057i	$0.542603\pi$
$810$	0	0
$811$	24.2815	0.852637	0.426319	−	0.904573i	$-0.359810\pi$
$811$	24.2815	0.852637	0.426319	+	0.904573i	$0.359810\pi$
$812$	62.9673	2.20972
$813$	0	0
$814$	−2.73191	−0.0957535
$815$	−0.155016	−0.00542998
$816$	0	0
$817$	−5.16824	−0.180814
$818$	−3.31265	−0.115824
$819$	0	0
$820$	−1.00140	−0.0349703
$821$	42.3738	1.47886	0.739428	−	0.673236i	$-0.235096\pi$
$821$	42.3738	1.47886	0.739428	+	0.673236i	$0.235096\pi$
$822$	0	0
$823$	−5.03535	−0.175521	−0.0877606	−	0.996142i	$-0.527971\pi$
$823$	−5.03535	−0.175521	−0.0877606	+	0.996142i	$0.527971\pi$
$824$	0.666754	0.0232275
$825$	0	0
$826$	−16.1587	−0.562231
$827$	−5.16603	−0.179641	−0.0898203	−	0.995958i	$-0.528629\pi$
$827$	−5.16603	−0.179641	−0.0898203	+	0.995958i	$0.528629\pi$
$828$	0	0
$829$	−7.38888	−0.256626	−0.128313	−	0.991734i	$-0.540956\pi$
$829$	−7.38888	−0.256626	−0.128313	+	0.991734i	$0.540956\pi$
$830$	−0.185337	−0.00643313
$831$	0	0
$832$	17.6042	0.610315
$833$	63.4769	2.19934
$834$	0	0
$835$	3.27848	0.113456
$836$	−1.79809	−0.0621882
$837$	0	0
$838$	−9.18499	−0.317290
$839$	4.95816	0.171175	0.0855873	−	0.996331i	$-0.472723\pi$
$839$	4.95816	0.171175	0.0855873	+	0.996331i	$0.472723\pi$
$840$	0	0
$841$	45.5290	1.56997
$842$	12.2294	0.421454
$843$	0	0
$844$	7.53383	0.259325
$845$	0.0790735	0.00272021
$846$	0	0
$847$	3.90557	0.134197
$848$	−21.9809	−0.754827
$849$	0	0
$850$	−13.9428	−0.478234
$851$	−18.3013	−0.627361
$852$	0	0
$853$	−18.1153	−0.620256	−0.310128	−	0.950695i	$-0.600372\pi$
$853$	−18.1153	−0.620256	−0.310128	+	0.950695i	$0.600372\pi$
$854$	−1.42148	−0.0486421
$855$	0	0
$856$	6.55638	0.224093
$857$	−16.3807	−0.559555	−0.279778	−	0.960065i	$-0.590261\pi$
$857$	−16.3807	−0.559555	−0.279778	+	0.960065i	$0.590261\pi$
$858$	0	0
$859$	−3.53744	−0.120696	−0.0603479	−	0.998177i	$-0.519221\pi$
$859$	−3.53744	−0.120696	−0.0603479	+	0.998177i	$0.519221\pi$
$860$	1.38226	0.0471346
$861$	0	0
$862$	−5.89106	−0.200650
$863$	−53.3494	−1.81604	−0.908018	−	0.418931i	$-0.862405\pi$
$863$	−53.3494	−1.81604	−0.908018	+	0.418931i	$0.862405\pi$
$864$	0	0
$865$	−0.238414	−0.00810633
$866$	−10.9226	−0.371165
$867$	0	0
$868$	67.3350	2.28550
$869$	9.20719	0.312333
$870$	0	0
$871$	−52.6669	−1.78455
$872$	−9.41533	−0.318843
$873$	0	0
$874$	0.854420	0.0289012
$875$	5.37502	0.181709
$876$	0	0
$877$	42.0727	1.42069	0.710347	−	0.703852i	$-0.248538\pi$
$877$	42.0727	1.42069	0.710347	+	0.703852i	$0.248538\pi$
$878$	−1.67277	−0.0564531
$879$	0	0
$880$	0.444372	0.0149798
$881$	−41.9114	−1.41203	−0.706015	−	0.708197i	$-0.749509\pi$
$881$	−41.9114	−1.41203	−0.706015	+	0.708197i	$0.749509\pi$
$882$	0	0
$883$	−42.7714	−1.43937	−0.719687	−	0.694299i	$-0.755715\pi$
$883$	−42.7714	−1.43937	−0.719687	+	0.694299i	$0.755715\pi$
$884$	50.6315	1.70292
$885$	0	0
$886$	−8.41015	−0.282544
$887$	−5.46844	−0.183612	−0.0918062	−	0.995777i	$-0.529264\pi$
$887$	−5.46844	−0.183612	−0.0918062	+	0.995777i	$0.529264\pi$
$888$	0	0
$889$	−33.2837	−1.11630
$890$	−0.799417	−0.0267965
$891$	0	0
$892$	17.3745	0.581741
$893$	−4.11602	−0.137738
$894$	0	0
$895$	−1.32109	−0.0441592
$896$	−38.2513	−1.27789
$897$	0	0
$898$	−0.848893	−0.0283279
$899$	79.6986	2.65810
$900$	0	0
$901$	−52.4565	−1.74758
$902$	1.41538	0.0471269
$903$	0	0
$904$	17.2953	0.575234
$905$	1.01127	0.0336158
$906$	0	0
$907$	−38.7228	−1.28577	−0.642885	−	0.765963i	$-0.722263\pi$
$907$	−38.7228	−1.28577	−0.642885	+	0.765963i	$0.722263\pi$
$908$	36.7750	1.22042
$909$	0	0
$910$	0.690936	0.0229043
$911$	28.0534	0.929451	0.464726	−	0.885455i	$-0.346153\pi$
$911$	28.0534	0.929451	0.464726	+	0.885455i	$0.346153\pi$
$912$	0	0
$913$	−3.69303	−0.122222
$914$	4.51103	0.149212
$915$	0	0
$916$	−26.4615	−0.874314
$917$	−48.1604	−1.59040
$918$	0	0
$919$	25.5560	0.843013	0.421507	−	0.906825i	$-0.361501\pi$
$919$	25.5560	0.843013	0.421507	+	0.906825i	$0.361501\pi$
$920$	−0.473243	−0.0156024
$921$	0	0
$922$	−2.15725	−0.0710452
$923$	11.8001	0.388405
$924$	0	0
$925$	−37.3875	−1.22929
$926$	4.05996	0.133418
$927$	0	0
$928$	−34.4304	−1.13023
$929$	45.0809	1.47906	0.739529	−	0.673125i	$-0.235048\pi$
$929$	45.0809	1.47906	0.739529	+	0.673125i	$0.235048\pi$
$930$	0	0
$931$	−7.94659	−0.260439
$932$	17.3607	0.568669
$933$	0	0
$934$	7.71127	0.252320
$935$	1.06047	0.0346812
$936$	0	0
$937$	−49.6485	−1.62195	−0.810973	−	0.585083i	$-0.801062\pi$
$937$	−49.6485	−1.62195	−0.810973	+	0.585083i	$0.801062\pi$
$938$	21.2375	0.693429
$939$	0	0
$940$	1.10084	0.0359055
$941$	−14.5097	−0.473003	−0.236502	−	0.971631i	$-0.576001\pi$
$941$	−14.5097	−0.473003	−0.236502	+	0.971631i	$0.576001\pi$
$942$	0	0
$943$	9.48174	0.308768
$944$	−36.6344	−1.19235
$945$	0	0
$946$	−1.95369	−0.0635199
$947$	10.5780	0.343739	0.171870	−	0.985120i	$-0.445019\pi$
$947$	10.5780	0.343739	0.171870	+	0.985120i	$0.445019\pi$
$948$	0	0
$949$	−8.60259	−0.279252
$950$	1.74548	0.0566308
$951$	0	0
$952$	−42.2817	−1.37036
$953$	23.0486	0.746617	0.373308	−	0.927707i	$-0.378223\pi$
$953$	23.0486	0.746617	0.373308	+	0.927707i	$0.378223\pi$
$954$	0	0
$955$	1.91765	0.0620536
$956$	29.5347	0.955219
$957$	0	0
$958$	10.6792	0.345029
$959$	−27.8312	−0.898715
$960$	0	0
$961$	54.2268	1.74925
$962$	−9.63034	−0.310495
$963$	0	0
$964$	33.5190	1.07957
$965$	2.76527	0.0890173
$966$	0	0
$967$	−11.6095	−0.373335	−0.186668	−	0.982423i	$-0.559769\pi$
$967$	−11.6095	−0.373335	−0.186668	+	0.982423i	$0.559769\pi$
$968$	−1.40764	−0.0452431
$969$	0	0
$970$	0.767524	0.0246437
$971$	−23.5383	−0.755380	−0.377690	−	0.925932i	$-0.623282\pi$
$971$	−23.5383	−0.755380	−0.377690	+	0.925932i	$0.623282\pi$
$972$	0	0
$973$	65.6548	2.10480
$974$	12.3091	0.394410
$975$	0	0
$976$	−3.22274	−0.103157
$977$	−39.8465	−1.27480	−0.637402	−	0.770532i	$-0.719991\pi$
$977$	−39.8465	−1.27480	−0.637402	+	0.770532i	$0.719991\pi$
$978$	0	0
$979$	−15.9292	−0.509101
$980$	2.12534	0.0678914
$981$	0	0
$982$	11.4491	0.365356
$983$	48.2392	1.53859	0.769296	−	0.638892i	$-0.220607\pi$
$983$	48.2392	1.53859	0.769296	+	0.638892i	$0.220607\pi$
$984$	0	0
$985$	−1.58476	−0.0504948
$986$	−24.1656	−0.769589
$987$	0	0
$988$	−6.33849	−0.201654
$989$	−13.0879	−0.416173
$990$	0	0
$991$	−57.7302	−1.83386	−0.916930	−	0.399048i	$-0.869341\pi$
$991$	−57.7302	−1.83386	−0.916930	+	0.399048i	$0.869341\pi$
$992$	−36.8186	−1.16899
$993$	0	0
$994$	−4.75830	−0.150924
$995$	−1.19254	−0.0378061
$996$	0	0
$997$	28.4504	0.901033	0.450516	−	0.892768i	$-0.351240\pi$
$997$	28.4504	0.901033	0.450516	+	0.892768i	$0.351240\pi$
$998$	−11.0731	−0.350513
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.16	✓	25	1.1	even	1	trivial
6039.2.a.o.1.10	yes	25	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+0.363962 q^{2} -1.86753 q^{4} -0.137887 q^{5} +3.90557 q^{7} -1.40764 q^{8} +O(q^{10})\)	\(q+0.363962 q^{2} -1.86753 q^{4} -0.137887 q^{5} +3.90557 q^{7} -1.40764 q^{8} -0.0501855 q^{10} -1.00000 q^{11} -3.52513 q^{13} +1.42148 q^{14} +3.22274 q^{16} +7.69092 q^{17} -0.962815 q^{19} +0.257508 q^{20} -0.363962 q^{22} -2.43822 q^{23} -4.98099 q^{25} -1.28301 q^{26} -7.29378 q^{28} -8.63302 q^{29} -9.23184 q^{31} +3.98822 q^{32} +2.79920 q^{34} -0.538526 q^{35} +7.50603 q^{37} -0.350428 q^{38} +0.194094 q^{40} -3.88880 q^{41} +5.36784 q^{43} +1.86753 q^{44} -0.887418 q^{46} +4.27499 q^{47} +8.25349 q^{49} -1.81289 q^{50} +6.58329 q^{52} -6.82057 q^{53} +0.137887 q^{55} -5.49762 q^{56} -3.14209 q^{58} -11.3675 q^{59} -1.00000 q^{61} -3.36004 q^{62} -4.99391 q^{64} +0.486068 q^{65} +14.9404 q^{67} -14.3630 q^{68} -0.196003 q^{70} -3.34743 q^{71} +2.44036 q^{73} +2.73191 q^{74} +1.79809 q^{76} -3.90557 q^{77} -9.20719 q^{79} -0.444372 q^{80} -1.41538 q^{82} +3.69303 q^{83} -1.06047 q^{85} +1.95369 q^{86} +1.40764 q^{88} +15.9292 q^{89} -13.7676 q^{91} +4.55344 q^{92} +1.55593 q^{94} +0.132759 q^{95} -15.2937 q^{97} +3.00396 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8}+O(q^{10}) \) 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8	\( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8} - 25 q^{11} + 4 q^{13} - 18 q^{14} + 21 q^{16} - 20 q^{17} + 14 q^{19} - 12 q^{20} + 5 q^{22} - 20 q^{23} + 13 q^{25} - 16 q^{26} - 14 q^{28} - 28 q^{29} - 12 q^{31} - 35 q^{32} + 6 q^{34} - 10 q^{35} - 8 q^{37} - 32 q^{38} + 24 q^{40} - 26 q^{41} + 18 q^{43} - 25 q^{44} + 4 q^{46} - 12 q^{47} + 23 q^{49} - 43 q^{50} + 22 q^{52} - 36 q^{53} + 4 q^{55} - 26 q^{56} - 20 q^{58} - 46 q^{59} - 25 q^{61} + 14 q^{62} - 13 q^{64} - 60 q^{65} - 20 q^{67} - 44 q^{68} - 20 q^{70} - 52 q^{71} + 6 q^{73} - 32 q^{74} - 4 q^{77} + 26 q^{79} - 52 q^{80} + 6 q^{82} - 38 q^{83} - 4 q^{85} - 34 q^{86} + 15 q^{88} - 82 q^{89} - 58 q^{91} - 36 q^{92} + 16 q^{94} - 30 q^{95} - 35 q^{98}+O(q^{100}) \) 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8 - 25 * q^11 + 4 * q^13 - 18 * q^14 + 21 * q^16 - 20 * q^17 + 14 * q^19 - 12 * q^20 + 5 * q^22 - 20 * q^23 + 13 * q^25 - 16 * q^26 - 14 * q^28 - 28 * q^29 - 12 * q^31 - 35 * q^32 + 6 * q^34 - 10 * q^35 - 8 * q^37 - 32 * q^38 + 24 * q^40 - 26 * q^41 + 18 * q^43 - 25 * q^44 + 4 * q^46 - 12 * q^47 + 23 * q^49 - 43 * q^50 + 22 * q^52 - 36 * q^53 + 4 * q^55 - 26 * q^56 - 20 * q^58 - 46 * q^59 - 25 * q^61 + 14 * q^62 - 13 * q^64 - 60 * q^65 - 20 * q^67 - 44 * q^68 - 20 * q^70 - 52 * q^71 + 6 * q^73 - 32 * q^74 - 4 * q^77 + 26 * q^79 - 52 * q^80 + 6 * q^82 - 38 * q^83 - 4 * q^85 - 34 * q^86 + 15 * q^88 - 82 * q^89 - 58 * q^91 - 36 * q^92 + 16 * q^94 - 30 * q^95 - 35 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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