LMFDB - Embedded newform 6039.2.a.o.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)

$f(q)$	$=$	$q-0.363962 q^{2} -1.86753 q^{4} +0.137887 q^{5} +3.90557 q^{7} +1.40764 q^{8} +O(q^{10})$	\(q-0.363962 q^{2} -1.86753 q^{4} +0.137887 q^{5} +3.90557 q^{7} +1.40764 q^{8} -0.0501855 q^{10} +1.00000 q^{11} -3.52513 q^{13} -1.42148 q^{14} +3.22274 q^{16} -7.69092 q^{17} -0.962815 q^{19} -0.257508 q^{20} -0.363962 q^{22} +2.43822 q^{23} -4.98099 q^{25} +1.28301 q^{26} -7.29378 q^{28} +8.63302 q^{29} -9.23184 q^{31} -3.98822 q^{32} +2.79920 q^{34} +0.538526 q^{35} +7.50603 q^{37} +0.350428 q^{38} +0.194094 q^{40} +3.88880 q^{41} +5.36784 q^{43} -1.86753 q^{44} -0.887418 q^{46} -4.27499 q^{47} +8.25349 q^{49} +1.81289 q^{50} +6.58329 q^{52} +6.82057 q^{53} +0.137887 q^{55} +5.49762 q^{56} -3.14209 q^{58} +11.3675 q^{59} -1.00000 q^{61} +3.36004 q^{62} -4.99391 q^{64} -0.486068 q^{65} +14.9404 q^{67} +14.3630 q^{68} -0.196003 q^{70} +3.34743 q^{71} +2.44036 q^{73} -2.73191 q^{74} +1.79809 q^{76} +3.90557 q^{77} -9.20719 q^{79} +0.444372 q^{80} -1.41538 q^{82} -3.69303 q^{83} -1.06047 q^{85} -1.95369 q^{86} +1.40764 q^{88} -15.9292 q^{89} -13.7676 q^{91} -4.55344 q^{92} +1.55593 q^{94} -0.132759 q^{95} -15.2937 q^{97} -3.00396 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.541074\pi$
$2$	−0.363962	−0.257360	−0.128680	+	0.991686i	$0.541074\pi$
$3$	0	0
$3$	0	0
$4$	−1.86753	−0.933766
$5$	0.137887	0.0616648	0.0308324	−	0.999525i	$-0.490184\pi$
$5$	0.137887	0.0616648	0.0308324	+	0.999525i	$0.490184\pi$
$6$	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	$-0.882519\pi$
$17$	−7.69092	−1.86532	−0.932661	+	0.360754i	$0.882519\pi$
$18$	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	$-0.418187\pi$
$23$	2.43822	0.508403	0.254202	+	0.967151i	$0.418187\pi$
$24$	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	$-0.204006\pi$
$29$	8.63302	1.60311	0.801556	+	0.597920i	$0.204006\pi$
$30$	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	$-0.401790\pi$
$41$	3.88880	0.607329	0.303665	+	0.952779i	$0.401790\pi$
$42$	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	$-0.600927\pi$
$47$	−4.27499	−0.623571	−0.311786	+	0.950152i	$0.600927\pi$
$48$	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	$-0.344817\pi$
$53$	6.82057	0.936877	0.468439	+	0.883496i	$0.344817\pi$
$54$	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	$-0.234844\pi$
$59$	11.3675	1.47992	0.739960	+	0.672651i	$0.234844\pi$
$60$	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	$-0.436350\pi$
$71$	3.34743	0.397266	0.198633	+	0.980074i	$0.436350\pi$
$72$	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	$-0.564966\pi$
$83$	−3.69303	−0.405363	−0.202681	+	0.979245i	$0.564966\pi$
$84$	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	$-0.819953\pi$
$89$	−15.9292	−1.68850	−0.844248	+	0.535953i	$0.819953\pi$
$90$	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	$-0.510396\pi$
$101$	−0.656362	−0.0653104	−0.0326552	+	0.999467i	$0.510396\pi$
$102$	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	$-0.427716\pi$
$107$	4.65773	0.450280	0.225140	+	0.974326i	$0.427716\pi$
$108$	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	$-0.303864\pi$
$113$	12.2868	1.15585	0.577923	+	0.816092i	$0.303864\pi$
$114$	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	$-0.318919\pi$
$131$	12.3312	1.07738	0.538691	+	0.842504i	$0.318919\pi$
$132$	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	$-0.401541\pi$
$137$	7.12602	0.608817	0.304408	+	0.952542i	$0.401541\pi$
$138$	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	$-0.442579\pi$
$149$	4.38013	0.358834	0.179417	+	0.983773i	$0.442579\pi$
$150$	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	$-0.128233\pi$
$167$	23.7766	1.83989	0.919945	+	0.392047i	$0.128233\pi$
$168$	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	$-0.520937\pi$
$173$	−1.72906	−0.131458	−0.0657290	+	0.997838i	$0.520937\pi$
$174$	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	$-0.616561\pi$
$179$	−9.58099	−0.716117	−0.358058	+	0.933699i	$0.616561\pi$
$180$	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	$-0.332173\pi$
$191$	13.9074	1.00631	0.503153	+	0.864197i	$0.332173\pi$
$192$	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	$-0.634272\pi$
$197$	−11.4932	−0.818859	−0.409430	+	0.912342i	$0.634272\pi$
$198$	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	$-0.273302\pi$
$227$	19.6918	1.30699	0.653495	+	0.756931i	$0.273302\pi$
$228$	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	$-0.401510\pi$
$233$	9.29608	0.609006	0.304503	+	0.952511i	$0.401510\pi$
$234$	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	$-0.329095\pi$
$239$	15.8148	1.02298	0.511488	+	0.859291i	$0.329095\pi$
$240$	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	$-0.224170\pi$
$251$	24.1477	1.52419	0.762096	+	0.647464i	$0.224170\pi$
$252$	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	$-0.293153\pi$
$257$	19.3994	1.21010	0.605050	+	0.796188i	$0.293153\pi$
$258$	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	$-0.353461\pi$
$263$	14.4099	0.888552	0.444276	+	0.895890i	$0.353461\pi$
$264$	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	$-0.380667\pi$
$269$	12.0114	0.732350	0.366175	+	0.930546i	$0.380667\pi$
$270$	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	$-0.251439\pi$
$281$	23.5991	1.40780	0.703902	+	0.710297i	$0.251439\pi$
$282$	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	$-0.830801\pi$
$293$	−29.5108	−1.72404	−0.862020	+	0.506874i	$0.830801\pi$
$294$	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	$-0.696667\pi$
$311$	−20.4315	−1.15856	−0.579281	+	0.815128i	$0.696667\pi$
$312$	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	$-0.532425\pi$
$317$	−3.62111	−0.203382	−0.101691	+	0.994816i	$0.532425\pi$
$318$	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	$-0.190240\pi$
$347$	30.7978	1.65331	0.826657	+	0.562706i	$0.190240\pi$
$348$	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	$-0.657689\pi$
$353$	−17.8631	−0.950758	−0.475379	+	0.879781i	$0.657689\pi$
$354$	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	$-0.583228\pi$
$359$	−9.79570	−0.516998	−0.258499	+	0.966012i	$0.583228\pi$
$360$	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	$-0.379029\pi$
$383$	14.5196	0.741918	0.370959	+	0.928649i	$0.379029\pi$
$384$	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	$-0.468964\pi$
$389$	3.84007	0.194699	0.0973495	+	0.995250i	$0.468964\pi$
$390$	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	$-0.224785\pi$
$401$	30.4718	1.52169	0.760844	+	0.648934i	$0.224785\pi$
$402$	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	$-0.288577\pi$
$419$	25.2361	1.23287	0.616433	+	0.787408i	$0.288577\pi$
$420$	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	$-0.372536\pi$
$431$	16.1859	0.779648	0.389824	+	0.920889i	$0.372536\pi$
$432$	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	$-0.315036\pi$
$443$	23.1072	1.09786	0.548928	+	0.835869i	$0.315036\pi$
$444$	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	$-0.482473\pi$
$449$	2.33237	0.110071	0.0550356	+	0.998484i	$0.482473\pi$
$450$	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	$-0.455924\pi$
$461$	5.92712	0.276054	0.138027	+	0.990428i	$0.455924\pi$
$462$	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	$-0.663080\pi$
$467$	−21.1870	−0.980418	−0.490209	+	0.871605i	$0.663080\pi$
$468$	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	$-0.733845\pi$
$479$	−29.3415	−1.34065	−0.670323	+	0.742069i	$0.733845\pi$
$480$	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	$-0.751221\pi$
$491$	−31.4569	−1.41963	−0.709815	+	0.704388i	$0.751221\pi$
$492$	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	$-0.220332\pi$
$503$	34.5318	1.53970	0.769848	+	0.638227i	$0.220332\pi$
$504$	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	$-0.315427\pi$
$509$	24.7224	1.09580	0.547900	+	0.836544i	$0.315427\pi$
$510$	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	$-0.739231\pi$
$521$	−31.1697	−1.36557	−0.682785	+	0.730619i	$0.739231\pi$
$522$	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	$-0.538811\pi$
$557$	−5.74103	−0.243255	−0.121628	+	0.992576i	$0.538811\pi$
$558$	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	$-0.464180\pi$
$563$	5.32893	0.224588	0.112294	+	0.993675i	$0.464180\pi$
$564$	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	$-0.473089\pi$
$569$	4.02852	0.168884	0.0844422	+	0.996428i	$0.473089\pi$
$570$	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	$-0.301613\pi$
$587$	28.2827	1.16735	0.583677	+	0.811986i	$0.301613\pi$
$588$	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	$-0.528008\pi$
$593$	−4.27991	−0.175755	−0.0878774	+	0.996131i	$0.528008\pi$
$594$	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	$-0.403406\pi$
$599$	14.6271	0.597648	0.298824	+	0.954308i	$0.403406\pi$
$600$	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	$-0.688768\pi$
$617$	−27.7645	−1.11776	−0.558878	+	0.829250i	$0.688768\pi$
$618$	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	$-0.470781\pi$
$641$	4.64156	0.183331	0.0916653	+	0.995790i	$0.470781\pi$
$642$	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	$-0.355109\pi$
$647$	22.3651	0.879264	0.439632	+	0.898178i	$0.355109\pi$
$648$	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	$-0.576348\pi$
$653$	−12.1413	−0.475125	−0.237562	+	0.971372i	$0.576348\pi$
$654$	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	$-0.440091\pi$
$659$	9.60613	0.374202	0.187101	+	0.982341i	$0.440091\pi$
$660$	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	$-0.233527\pi$
$677$	38.6509	1.48547	0.742737	+	0.669583i	$0.233527\pi$
$678$	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	$-0.866472\pi$
$683$	−47.7367	−1.82659	−0.913296	+	0.407296i	$0.866472\pi$
$684$	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	$-0.647842\pi$
$701$	−23.7197	−0.895879	−0.447940	+	0.894064i	$0.647842\pi$
$702$	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	$-0.732608\pi$
$719$	−35.7934	−1.33487	−0.667435	+	0.744668i	$0.732608\pi$
$720$	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	$-0.492725\pi$
$743$	1.24583	0.0457053	0.0228526	+	0.999739i	$0.492725\pi$
$744$	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	$-0.483755\pi$
$761$	2.81443	0.102023	0.0510116	+	0.998698i	$0.483755\pi$
$762$	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	$-0.552511\pi$
$773$	−9.13158	−0.328440	−0.164220	+	0.986424i	$0.552511\pi$
$774$	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	$-0.579893\pi$
$797$	−14.0233	−0.496731	−0.248366	+	0.968666i	$0.579893\pi$
$798$	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	$-0.457397\pi$
$809$	7.59101	0.266886	0.133443	+	0.991057i	$0.457397\pi$
$810$	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	$-0.764904\pi$
$821$	−42.3738	−1.47886	−0.739428	+	0.673236i	$0.764904\pi$
$822$	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	$-0.471371\pi$
$827$	5.16603	0.179641	0.0898203	+	0.995958i	$0.471371\pi$
$828$	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	$-0.527277\pi$
$839$	−4.95816	−0.171175	−0.0855873	+	0.996331i	$0.527277\pi$
$840$	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	$-0.409739\pi$
$857$	16.3807	0.559555	0.279778	+	0.960065i	$0.409739\pi$
$858$	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	$-0.137595\pi$
$863$	53.3494	1.81604	0.908018	+	0.418931i	$0.137595\pi$
$864$	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	$-0.250491\pi$
$881$	41.9114	1.41203	0.706015	+	0.708197i	$0.250491\pi$
$882$	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	$-0.470736\pi$
$887$	5.46844	0.183612	0.0918062	+	0.995777i	$0.470736\pi$
$888$	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	$-0.653847\pi$
$911$	−28.0534	−0.929451	−0.464726	+	0.885455i	$0.653847\pi$
$912$	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	$-0.764952\pi$
$929$	−45.0809	−1.47906	−0.739529	+	0.673125i	$0.764952\pi$
$930$	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	$-0.423999\pi$
$941$	14.5097	0.473003	0.236502	+	0.971631i	$0.423999\pi$
$942$	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	$-0.554981\pi$
$947$	−10.5780	−0.343739	−0.171870	+	0.985120i	$0.554981\pi$
$948$	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	$-0.621777\pi$
$953$	−23.0486	−0.746617	−0.373308	+	0.927707i	$0.621777\pi$
$954$	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	$-0.376718\pi$
$971$	23.5383	0.755380	0.377690	+	0.925932i	$0.376718\pi$
$972$	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	$-0.280009\pi$
$977$	39.8465	1.27480	0.637402	+	0.770532i	$0.280009\pi$
$978$	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	$-0.779393\pi$
$983$	−48.2392	−1.53859	−0.769296	+	0.638892i	$0.779393\pi$
$984$	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.o.1.10	yes	25
3.2	odd	2		6039.2.a.n.1.16	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.16	✓	25	3.2	odd	2
6039.2.a.o.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)

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