LMFDB - Embedded newform 4114.2.a.o.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4114,2,Mod(1,4114)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4114.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4114, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4114 = 2 \cdot 11^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4114.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,-1,3,-3,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.8504553916$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.69963$ of defining polynomial
Character	$\chi$	$=$	4114.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.69963 q^{3} +1.00000 q^{4} -1.11126 q^{5} -1.69963 q^{6} +1.58836 q^{7} -1.00000 q^{8} -0.111264 q^{9} +1.11126 q^{10} +1.69963 q^{12} +0.300372 q^{13} -1.58836 q^{14} -1.88874 q^{15} +1.00000 q^{16} -1.00000 q^{17} +0.111264 q^{18} +6.98762 q^{19} -1.11126 q^{20} +2.69963 q^{21} -3.17673 q^{23} -1.69963 q^{24} -3.76509 q^{25} -0.300372 q^{26} -5.28799 q^{27} +1.58836 q^{28} -8.57598 q^{29} +1.88874 q^{30} -10.5760 q^{31} -1.00000 q^{32} +1.00000 q^{34} -1.76509 q^{35} -0.111264 q^{36} -4.18911 q^{37} -6.98762 q^{38} +0.510520 q^{39} +1.11126 q^{40} +4.28799 q^{41} -2.69963 q^{42} -2.06546 q^{43} +0.123644 q^{45} +3.17673 q^{46} +1.11126 q^{47} +1.69963 q^{48} -4.47710 q^{49} +3.76509 q^{50} -1.69963 q^{51} +0.300372 q^{52} -7.62178 q^{53} +5.28799 q^{54} -1.58836 q^{56} +11.8764 q^{57} +8.57598 q^{58} +7.97524 q^{59} -1.88874 q^{60} -6.35346 q^{61} +10.5760 q^{62} -0.176728 q^{63} +1.00000 q^{64} -0.333792 q^{65} +12.5760 q^{67} -1.00000 q^{68} -5.39926 q^{69} +1.76509 q^{70} +2.57598 q^{71} +0.111264 q^{72} +9.47710 q^{73} +4.18911 q^{74} -6.39926 q^{75} +6.98762 q^{76} -0.510520 q^{78} +2.52290 q^{79} -1.11126 q^{80} -8.65383 q^{81} -4.28799 q^{82} -8.18911 q^{83} +2.69963 q^{84} +1.11126 q^{85} +2.06546 q^{86} -14.5760 q^{87} -1.58836 q^{89} -0.123644 q^{90} +0.477100 q^{91} -3.17673 q^{92} -17.9752 q^{93} -1.11126 q^{94} -7.76509 q^{95} -1.69963 q^{96} +2.00000 q^{97} +4.47710 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - q^{3} + 3 q^{4} - 3 q^{5} + q^{6} - q^{7} - 3 q^{8} + 3 q^{10} - q^{12} + 7 q^{13} + q^{14} - 6 q^{15} + 3 q^{16} - 3 q^{17} + 3 q^{19} - 3 q^{20} + 2 q^{21} + 2 q^{23} + q^{24} + 6 q^{25}+ \cdots + 8 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 - q^3 + 3 * q^4 - 3 * q^5 + q^6 - q^7 - 3 * q^8 + 3 * q^10 - q^12 + 7 * q^13 + q^14 - 6 * q^15 + 3 * q^16 - 3 * q^17 + 3 * q^19 - 3 * q^20 + 2 * q^21 + 2 * q^23 + q^24 + 6 * q^25 - 7 * q^26 - 4 * q^27 - q^28 - 2 * q^29 + 6 * q^30 - 8 * q^31 - 3 * q^32 + 3 * q^34 + 12 * q^35 - 19 * q^37 - 3 * q^38 - 11 * q^39 + 3 * q^40 + q^41 - 2 * q^42 + 5 * q^43 + 18 * q^45 - 2 * q^46 + 3 * q^47 - q^48 - 8 * q^49 - 6 * q^50 + q^51 + 7 * q^52 - 10 * q^53 + 4 * q^54 + q^56 + 18 * q^57 + 2 * q^58 - 12 * q^59 - 6 * q^60 + 4 * q^61 + 8 * q^62 + 11 * q^63 + 3 * q^64 + 14 * q^67 - 3 * q^68 - 4 * q^69 - 12 * q^70 - 16 * q^71 + 23 * q^73 + 19 * q^74 - 7 * q^75 + 3 * q^76 + 11 * q^78 + 13 * q^79 - 3 * q^80 - 9 * q^81 - q^82 - 31 * q^83 + 2 * q^84 + 3 * q^85 - 5 * q^86 - 20 * q^87 + q^89 - 18 * q^90 - 4 * q^91 + 2 * q^92 - 18 * q^93 - 3 * q^94 - 6 * q^95 + q^96 + 6 * q^97 + 8 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.69963	0.981281	0.490640	−	0.871362i	$-0.336763\pi$
$3$	1.69963	0.981281	0.490640	+	0.871362i	$0.336763\pi$
$4$	1.00000	0.500000
$5$	−1.11126	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$5$	−1.11126	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$6$	−1.69963	−0.693870
$7$	1.58836	0.600345	0.300173	−	0.953885i	$-0.402956\pi$
$7$	1.58836	0.600345	0.300173	+	0.953885i	$0.402956\pi$
$8$	−1.00000	−0.353553
$9$	−0.111264	−0.0370881
$10$	1.11126	0.351413
$11$	0	0
$11$	0	0
$12$	1.69963	0.490640
$13$	0.300372	0.0833082	0.0416541	−	0.999132i	$-0.486737\pi$
$13$	0.300372	0.0833082	0.0416541	+	0.999132i	$0.486737\pi$
$14$	−1.58836	−0.424508
$15$	−1.88874	−0.487669
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0.111264	0.0262252
$19$	6.98762	1.60307	0.801535	−	0.597948i	$-0.204017\pi$
$19$	6.98762	1.60307	0.801535	+	0.597948i	$0.204017\pi$
$20$	−1.11126	−0.248486
$21$	2.69963	0.589107
$22$	0	0
$23$	−3.17673	−0.662394	−0.331197	−	0.943562i	$-0.607452\pi$
$23$	−3.17673	−0.662394	−0.331197	+	0.943562i	$0.607452\pi$
$24$	−1.69963	−0.346935
$25$	−3.76509	−0.753018
$26$	−0.300372	−0.0589078
$27$	−5.28799	−1.01767
$28$	1.58836	0.300173
$29$	−8.57598	−1.59252	−0.796260	−	0.604954i	$-0.793191\pi$
$29$	−8.57598	−1.59252	−0.796260	+	0.604954i	$0.793191\pi$
$30$	1.88874	0.344834
$31$	−10.5760	−1.89950	−0.949751	−	0.313005i	$-0.898664\pi$
$31$	−10.5760	−1.89950	−0.949751	+	0.313005i	$0.898664\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	−1.76509	−0.298355
$36$	−0.111264	−0.0185440
$37$	−4.18911	−0.688685	−0.344343	−	0.938844i	$-0.611898\pi$
$37$	−4.18911	−0.688685	−0.344343	+	0.938844i	$0.611898\pi$
$38$	−6.98762	−1.13354
$39$	0.510520	0.0817487
$40$	1.11126	0.175706
$41$	4.28799	0.669672	0.334836	−	0.942276i	$-0.391319\pi$
$41$	4.28799	0.669672	0.334836	+	0.942276i	$0.391319\pi$
$42$	−2.69963	−0.416562
$43$	−2.06546	−0.314980	−0.157490	−	0.987521i	$-0.550340\pi$
$43$	−2.06546	−0.314980	−0.157490	+	0.987521i	$0.550340\pi$
$44$	0	0
$45$	0.123644	0.0184317
$46$	3.17673	0.468383
$47$	1.11126	0.162095	0.0810473	−	0.996710i	$-0.474174\pi$
$47$	1.11126	0.162095	0.0810473	+	0.996710i	$0.474174\pi$
$48$	1.69963	0.245320
$49$	−4.47710	−0.639586
$50$	3.76509	0.532464
$51$	−1.69963	−0.237996
$52$	0.300372	0.0416541
$53$	−7.62178	−1.04693	−0.523466	−	0.852046i	$-0.675361\pi$
$53$	−7.62178	−1.04693	−0.523466	+	0.852046i	$0.675361\pi$
$54$	5.28799	0.719605
$55$	0	0
$56$	−1.58836	−0.212254
$57$	11.8764	1.57306
$58$	8.57598	1.12608
$59$	7.97524	1.03829	0.519144	−	0.854687i	$-0.326251\pi$
$59$	7.97524	1.03829	0.519144	+	0.854687i	$0.326251\pi$
$60$	−1.88874	−0.243835
$61$	−6.35346	−0.813477	−0.406738	−	0.913545i	$-0.633334\pi$
$61$	−6.35346	−0.813477	−0.406738	+	0.913545i	$0.633334\pi$
$62$	10.5760	1.34315
$63$	−0.176728	−0.0222656
$64$	1.00000	0.125000
$65$	−0.333792	−0.0414019
$66$	0	0
$67$	12.5760	1.53640	0.768201	−	0.640209i	$-0.221152\pi$
$67$	12.5760	1.53640	0.768201	+	0.640209i	$0.221152\pi$
$68$	−1.00000	−0.121268
$69$	−5.39926	−0.649994
$70$	1.76509	0.210969
$71$	2.57598	0.305713	0.152857	−	0.988248i	$-0.451153\pi$
$71$	2.57598	0.305713	0.152857	+	0.988248i	$0.451153\pi$
$72$	0.111264	0.0131126
$73$	9.47710	1.10921	0.554605	−	0.832114i	$-0.312869\pi$
$73$	9.47710	1.10921	0.554605	+	0.832114i	$0.312869\pi$
$74$	4.18911	0.486974
$75$	−6.39926	−0.738922
$76$	6.98762	0.801535
$77$	0	0
$78$	−0.510520	−0.0578051
$79$	2.52290	0.283848	0.141924	−	0.989878i	$-0.454671\pi$
$79$	2.52290	0.283848	0.141924	+	0.989878i	$0.454671\pi$
$80$	−1.11126	−0.124243
$81$	−8.65383	−0.961536
$82$	−4.28799	−0.473530
$83$	−8.18911	−0.898871	−0.449436	−	0.893313i	$-0.648375\pi$
$83$	−8.18911	−0.898871	−0.449436	+	0.893313i	$0.648375\pi$
$84$	2.69963	0.294554
$85$	1.11126	0.120534
$86$	2.06546	0.222725
$87$	−14.5760	−1.56271
$88$	0	0
$89$	−1.58836	−0.168366	−0.0841831	−	0.996450i	$-0.526828\pi$
$89$	−1.58836	−0.168366	−0.0841831	+	0.996450i	$0.526828\pi$
$90$	−0.123644	−0.0130332
$91$	0.477100	0.0500137
$92$	−3.17673	−0.331197
$93$	−17.9752	−1.86395
$94$	−1.11126	−0.114618
$95$	−7.76509	−0.796682
$96$	−1.69963	−0.173468
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	4.47710	0.452255
$99$	0	0
$100$	−3.76509	−0.376509
$101$	−6.86398	−0.682991	−0.341496	−	0.939883i	$-0.610933\pi$
$101$	−6.86398	−0.682991	−0.341496	+	0.939883i	$0.610933\pi$
$102$	1.69963	0.168288
$103$	−0.522900	−0.0515229	−0.0257614	−	0.999668i	$-0.508201\pi$
$103$	−0.522900	−0.0515229	−0.0257614	+	0.999668i	$0.508201\pi$
$104$	−0.300372	−0.0294539
$105$	−3.00000	−0.292770
$106$	7.62178	0.740293
$107$	0.667585	0.0645379	0.0322689	−	0.999479i	$-0.489727\pi$
$107$	0.667585	0.0645379	0.0322689	+	0.999479i	$0.489727\pi$
$108$	−5.28799	−0.508837
$109$	9.53018	0.912826	0.456413	−	0.889768i	$-0.349134\pi$
$109$	9.53018	0.912826	0.456413	+	0.889768i	$0.349134\pi$
$110$	0	0
$111$	−7.11993	−0.675793
$112$	1.58836	0.150086
$113$	−13.6218	−1.28143	−0.640715	−	0.767779i	$-0.721362\pi$
$113$	−13.6218	−1.28143	−0.640715	+	0.767779i	$0.721362\pi$
$114$	−11.8764	−1.11232
$115$	3.53018	0.329191
$116$	−8.57598	−0.796260
$117$	−0.0334206	−0.00308974
$118$	−7.97524	−0.734180
$119$	−1.58836	−0.145605
$120$	1.88874	0.172417
$121$	0	0
$122$	6.35346	0.575215
$123$	7.28799	0.657136
$124$	−10.5760	−0.949751
$125$	9.74033	0.871202
$126$	0.176728	0.0157442
$127$	−3.84431	−0.341128	−0.170564	−	0.985347i	$-0.554559\pi$
$127$	−3.84431	−0.341128	−0.170564	+	0.985347i	$0.554559\pi$
$128$	−1.00000	−0.0883883
$129$	−3.51052	−0.309084
$130$	0.333792	0.0292755
$131$	−19.7280	−1.72364	−0.861820	−	0.507215i	$-0.830675\pi$
$131$	−19.7280	−1.72364	−0.861820	+	0.507215i	$0.830675\pi$
$132$	0	0
$133$	11.0989	0.962395
$134$	−12.5760	−1.08640
$135$	5.87636	0.505756
$136$	1.00000	0.0857493
$137$	−7.32141	−0.625511	−0.312755	−	0.949834i	$-0.601252\pi$
$137$	−7.32141	−0.625511	−0.312755	+	0.949834i	$0.601252\pi$
$138$	5.39926	0.459615
$139$	10.1309	0.859294	0.429647	−	0.902997i	$-0.358638\pi$
$139$	10.1309	0.859294	0.429647	+	0.902997i	$0.358638\pi$
$140$	−1.76509	−0.149177
$141$	1.88874	0.159060
$142$	−2.57598	−0.216172
$143$	0	0
$144$	−0.111264	−0.00927201
$145$	9.53018	0.791439
$146$	−9.47710	−0.784330
$147$	−7.60940	−0.627613
$148$	−4.18911	−0.344343
$149$	−4.14468	−0.339546	−0.169773	−	0.985483i	$-0.554303\pi$
$149$	−4.14468	−0.339546	−0.169773	+	0.985483i	$0.554303\pi$
$150$	6.39926	0.522497
$151$	13.9752	1.13729	0.568644	−	0.822583i	$-0.307468\pi$
$151$	13.9752	1.13729	0.568644	+	0.822583i	$0.307468\pi$
$152$	−6.98762	−0.566771
$153$	0.111264	0.00899517
$154$	0	0
$155$	11.7527	0.944001
$156$	0.510520	0.0408743
$157$	−16.5760	−1.32291	−0.661454	−	0.749986i	$-0.730060\pi$
$157$	−16.5760	−1.32291	−0.661454	+	0.749986i	$0.730060\pi$
$158$	−2.52290	−0.200711
$159$	−12.9542	−1.02733
$160$	1.11126	0.0878531
$161$	−5.04580	−0.397665
$162$	8.65383	0.679909
$163$	13.9098	1.08950	0.544749	−	0.838599i	$-0.316625\pi$
$163$	13.9098	1.08950	0.544749	+	0.838599i	$0.316625\pi$
$164$	4.28799	0.334836
$165$	0	0
$166$	8.18911	0.635598
$167$	13.5316	1.04710	0.523552	−	0.851994i	$-0.324607\pi$
$167$	13.5316	1.04710	0.523552	+	0.851994i	$0.324607\pi$
$168$	−2.69963	−0.208281
$169$	−12.9098	−0.993060
$170$	−1.11126	−0.0852301
$171$	−0.777472	−0.0594547
$172$	−2.06546	−0.157490
$173$	−15.2436	−1.15895	−0.579474	−	0.814991i	$-0.696742\pi$
$173$	−15.2436	−1.15895	−0.579474	+	0.814991i	$0.696742\pi$
$174$	14.5760	1.10500
$175$	−5.98034	−0.452071
$176$	0	0
$177$	13.5549	1.01885
$178$	1.58836	0.119053
$179$	−20.2225	−1.51150	−0.755751	−	0.654859i	$-0.772728\pi$
$179$	−20.2225	−1.51150	−0.755751	+	0.654859i	$0.772728\pi$
$180$	0.123644	0.00921587
$181$	−11.0073	−0.818165	−0.409082	−	0.912497i	$-0.634151\pi$
$181$	−11.0073	−0.818165	−0.409082	+	0.912497i	$0.634151\pi$
$182$	−0.477100	−0.0353650
$183$	−10.7985	−0.798249
$184$	3.17673	0.234191
$185$	4.65521	0.342257
$186$	17.9752	1.31801
$187$	0	0
$188$	1.11126	0.0810473
$189$	−8.39926	−0.610956
$190$	7.76509	0.563339
$191$	−20.6291	−1.49267	−0.746334	−	0.665572i	$-0.768188\pi$
$191$	−20.6291	−1.49267	−0.746334	+	0.665572i	$0.768188\pi$
$192$	1.69963	0.122660
$193$	3.12364	0.224845	0.112422	−	0.993661i	$-0.464139\pi$
$193$	3.12364	0.224845	0.112422	+	0.993661i	$0.464139\pi$
$194$	−2.00000	−0.143592
$195$	−0.567323	−0.0406268
$196$	−4.47710	−0.319793
$197$	−8.57598	−0.611014	−0.305507	−	0.952190i	$-0.598826\pi$
$197$	−8.57598	−0.611014	−0.305507	+	0.952190i	$0.598826\pi$
$198$	0	0
$199$	8.19777	0.581124	0.290562	−	0.956856i	$-0.406158\pi$
$199$	8.19777	0.581124	0.290562	+	0.956856i	$0.406158\pi$
$200$	3.76509	0.266232
$201$	21.3745	1.50764
$202$	6.86398	0.482948
$203$	−13.6218	−0.956062
$204$	−1.69963	−0.118998
$205$	−4.76509	−0.332808
$206$	0.522900	0.0364322
$207$	0.353456	0.0245669
$208$	0.300372	0.0208270
$209$	0	0
$210$	3.00000	0.207020
$211$	15.7775	1.08617	0.543083	−	0.839679i	$-0.317257\pi$
$211$	15.7775	1.08617	0.543083	+	0.839679i	$0.317257\pi$
$212$	−7.62178	−0.523466
$213$	4.37822	0.299990
$214$	−0.667585	−0.0456352
$215$	2.29528	0.156537
$216$	5.28799	0.359802
$217$	−16.7985	−1.14036
$218$	−9.53018	−0.645466
$219$	16.1075	1.08845
$220$	0	0
$221$	−0.300372	−0.0202052
$222$	7.11993	0.477858
$223$	−1.81089	−0.121266	−0.0606332	−	0.998160i	$-0.519312\pi$
$223$	−1.81089	−0.121266	−0.0606332	+	0.998160i	$0.519312\pi$
$224$	−1.58836	−0.106127
$225$	0.418920	0.0279280
$226$	13.6218	0.906108
$227$	3.24357	0.215283	0.107642	−	0.994190i	$-0.465670\pi$
$227$	3.24357	0.215283	0.107642	+	0.994190i	$0.465670\pi$
$228$	11.8764	0.786531
$229$	−2.95420	−0.195219	−0.0976095	−	0.995225i	$-0.531120\pi$
$229$	−2.95420	−0.195219	−0.0976095	+	0.995225i	$0.531120\pi$
$230$	−3.53018	−0.232773
$231$	0	0
$232$	8.57598	0.563041
$233$	9.47710	0.620865	0.310433	−	0.950595i	$-0.399526\pi$
$233$	9.47710	0.620865	0.310433	+	0.950595i	$0.399526\pi$
$234$	0.0334206	0.00218477
$235$	−1.23491	−0.0805565
$236$	7.97524	0.519144
$237$	4.28799	0.278535
$238$	1.58836	0.102958
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	−1.88874	−0.121917
$241$	−7.90840	−0.509425	−0.254712	−	0.967017i	$-0.581981\pi$
$241$	−7.90840	−0.509425	−0.254712	+	0.967017i	$0.581981\pi$
$242$	0	0
$243$	1.15569	0.0741374
$244$	−6.35346	−0.406738
$245$	4.97524	0.317856
$246$	−7.28799	−0.464665
$247$	2.09888	0.133549
$248$	10.5760	0.671576
$249$	−13.9184	−0.882045
$250$	−9.74033	−0.616033
$251$	18.1730	1.14707	0.573535	−	0.819181i	$-0.305572\pi$
$251$	18.1730	1.14707	0.573535	+	0.819181i	$0.305572\pi$
$252$	−0.176728	−0.0111328
$253$	0	0
$254$	3.84431	0.241214
$255$	1.88874	0.118277
$256$	1.00000	0.0625000
$257$	−29.4720	−1.83841	−0.919207	−	0.393776i	$-0.871169\pi$
$257$	−29.4720	−1.83841	−0.919207	+	0.393776i	$0.871169\pi$
$258$	3.51052	0.218555
$259$	−6.65383	−0.413449
$260$	−0.333792	−0.0207009
$261$	0.954200	0.0590635
$262$	19.7280	1.21880
$263$	8.57598	0.528818	0.264409	−	0.964411i	$-0.414823\pi$
$263$	8.57598	0.528818	0.264409	+	0.964411i	$0.414823\pi$
$264$	0	0
$265$	8.46982	0.520297
$266$	−11.0989	−0.680516
$267$	−2.69963	−0.165215
$268$	12.5760	0.768201
$269$	12.1396	0.740164	0.370082	−	0.928999i	$-0.379330\pi$
$269$	12.1396	0.740164	0.370082	+	0.928999i	$0.379330\pi$
$270$	−5.87636	−0.357624
$271$	−14.5091	−0.881368	−0.440684	−	0.897662i	$-0.645264\pi$
$271$	−14.5091	−0.881368	−0.440684	+	0.897662i	$0.645264\pi$
$272$	−1.00000	−0.0606339
$273$	0.810892	0.0490774
$274$	7.32141	0.442303
$275$	0	0
$276$	−5.39926	−0.324997
$277$	−29.5723	−1.77683	−0.888413	−	0.459046i	$-0.848191\pi$
$277$	−29.5723	−1.77683	−0.888413	+	0.459046i	$0.848191\pi$
$278$	−10.1309	−0.607613
$279$	1.17673	0.0704489
$280$	1.76509	0.105484
$281$	−31.4413	−1.87563	−0.937816	−	0.347131i	$-0.887156\pi$
$281$	−31.4413	−1.87563	−0.937816	+	0.347131i	$0.887156\pi$
$282$	−1.88874	−0.112473
$283$	15.1767	0.902163	0.451081	−	0.892483i	$-0.351038\pi$
$283$	15.1767	0.902163	0.451081	+	0.892483i	$0.351038\pi$
$284$	2.57598	0.152857
$285$	−13.1978	−0.781768
$286$	0	0
$287$	6.81089	0.402034
$288$	0.111264	0.00655630
$289$	1.00000	0.0588235
$290$	−9.53018	−0.559632
$291$	3.39926	0.199268
$292$	9.47710	0.554605
$293$	−3.62041	−0.211506	−0.105753	−	0.994392i	$-0.533725\pi$
$293$	−3.62041	−0.211506	−0.105753	+	0.994392i	$0.533725\pi$
$294$	7.60940	0.443790
$295$	−8.86260	−0.516000
$296$	4.18911	0.243487
$297$	0	0
$298$	4.14468	0.240095
$299$	−0.954200	−0.0551828
$300$	−6.39926	−0.369461
$301$	−3.28071	−0.189097
$302$	−13.9752	−0.804185
$303$	−11.6662	−0.670206
$304$	6.98762	0.400768
$305$	7.06037	0.404275
$306$	−0.111264	−0.00636055
$307$	−33.2967	−1.90034	−0.950170	−	0.311732i	$-0.899091\pi$
$307$	−33.2967	−1.90034	−0.950170	+	0.311732i	$0.899091\pi$
$308$	0	0
$309$	−0.888736	−0.0505584
$310$	−11.7527	−0.667509
$311$	−7.86907	−0.446214	−0.223107	−	0.974794i	$-0.571620\pi$
$311$	−7.86907	−0.446214	−0.223107	+	0.974794i	$0.571620\pi$
$312$	−0.510520	−0.0289025
$313$	23.4413	1.32498	0.662491	−	0.749070i	$-0.269499\pi$
$313$	23.4413	1.32498	0.662491	+	0.749070i	$0.269499\pi$
$314$	16.5760	0.935437
$315$	0.196391	0.0110654
$316$	2.52290	0.141924
$317$	−2.89011	−0.162325	−0.0811625	−	0.996701i	$-0.525863\pi$
$317$	−2.89011	−0.162325	−0.0811625	+	0.996701i	$0.525863\pi$
$318$	12.9542	0.726435
$319$	0	0
$320$	−1.11126	−0.0621216
$321$	1.13465	0.0633298
$322$	5.04580	0.281191
$323$	−6.98762	−0.388802
$324$	−8.65383	−0.480768
$325$	−1.13093	−0.0627326
$326$	−13.9098	−0.770391
$327$	16.1978	0.895739
$328$	−4.28799	−0.236765
$329$	1.76509	0.0973127
$330$	0	0
$331$	10.2894	0.565555	0.282777	−	0.959186i	$-0.408744\pi$
$331$	10.2894	0.565555	0.282777	+	0.959186i	$0.408744\pi$
$332$	−8.18911	−0.449436
$333$	0.466098	0.0255420
$334$	−13.5316	−0.740414
$335$	−13.9752	−0.763549
$336$	2.69963	0.147277
$337$	19.7120	1.07378	0.536891	−	0.843652i	$-0.319599\pi$
$337$	19.7120	1.07378	0.536891	+	0.843652i	$0.319599\pi$
$338$	12.9098	0.702199
$339$	−23.1520	−1.25744
$340$	1.11126	0.0602668
$341$	0	0
$342$	0.777472	0.0420409
$343$	−18.2298	−0.984317
$344$	2.06546	0.111362
$345$	6.00000	0.323029
$346$	15.2436	0.819499
$347$	29.8196	1.60080	0.800399	−	0.599468i	$-0.204621\pi$
$347$	29.8196	1.60080	0.800399	+	0.599468i	$0.204621\pi$
$348$	−14.5760	−0.781355
$349$	−33.2436	−1.77949	−0.889744	−	0.456460i	$-0.849117\pi$
$349$	−33.2436	−1.77949	−0.889744	+	0.456460i	$0.849117\pi$
$350$	5.98034	0.319662
$351$	−1.58836	−0.0847806
$352$	0	0
$353$	−2.66621	−0.141908	−0.0709540	−	0.997480i	$-0.522604\pi$
$353$	−2.66621	−0.141908	−0.0709540	+	0.997480i	$0.522604\pi$
$354$	−13.5549	−0.720437
$355$	−2.86260	−0.151931
$356$	−1.58836	−0.0841831
$357$	−2.69963	−0.142879
$358$	20.2225	1.06879
$359$	16.8654	0.890119	0.445060	−	0.895501i	$-0.353182\pi$
$359$	16.8654	0.890119	0.445060	+	0.895501i	$0.353182\pi$
$360$	−0.123644	−0.00651660
$361$	29.8268	1.56983
$362$	11.0073	0.578530
$363$	0	0
$364$	0.477100	0.0250068
$365$	−10.5316	−0.551247
$366$	10.7985	0.564447
$367$	14.9542	0.780603	0.390301	−	0.920687i	$-0.372371\pi$
$367$	14.9542	0.780603	0.390301	+	0.920687i	$0.372371\pi$
$368$	−3.17673	−0.165598
$369$	−0.477100	−0.0248368
$370$	−4.65521	−0.242013
$371$	−12.1062	−0.628521
$372$	−17.9752	−0.931973
$373$	−7.52152	−0.389450	−0.194725	−	0.980858i	$-0.562381\pi$
$373$	−7.52152	−0.389450	−0.194725	+	0.980858i	$0.562381\pi$
$374$	0	0
$375$	16.5549	0.854894
$376$	−1.11126	−0.0573091
$377$	−2.57598	−0.132670
$378$	8.39926	0.432011
$379$	−8.57461	−0.440448	−0.220224	−	0.975449i	$-0.570679\pi$
$379$	−8.57461	−0.440448	−0.220224	+	0.975449i	$0.570679\pi$
$380$	−7.76509	−0.398341
$381$	−6.53390	−0.334742
$382$	20.6291	1.05547
$383$	32.2619	1.64850	0.824252	−	0.566223i	$-0.191596\pi$
$383$	32.2619	1.64850	0.824252	+	0.566223i	$0.191596\pi$
$384$	−1.69963	−0.0867338
$385$	0	0
$386$	−3.12364	−0.158989
$387$	0.229812	0.0116820
$388$	2.00000	0.101535
$389$	17.5054	0.887560	0.443780	−	0.896136i	$-0.353637\pi$
$389$	17.5054	0.887560	0.443780	+	0.896136i	$0.353637\pi$
$390$	0.567323	0.0287275
$391$	3.17673	0.160654
$392$	4.47710	0.226128
$393$	−33.5302	−1.69137
$394$	8.57598	0.432052
$395$	−2.80361	−0.141065
$396$	0	0
$397$	−7.81089	−0.392017	−0.196009	−	0.980602i	$-0.562798\pi$
$397$	−7.81089	−0.392017	−0.196009	+	0.980602i	$0.562798\pi$
$398$	−8.19777	−0.410917
$399$	18.8640	0.944380
$400$	−3.76509	−0.188255
$401$	−9.10989	−0.454926	−0.227463	−	0.973787i	$-0.573043\pi$
$401$	−9.10989	−0.454926	−0.227463	+	0.973787i	$0.573043\pi$
$402$	−21.3745	−1.06606
$403$	−3.17673	−0.158244
$404$	−6.86398	−0.341496
$405$	9.61669	0.477857
$406$	13.6218	0.676038
$407$	0	0
$408$	1.69963	0.0841441
$409$	12.1730	0.601917	0.300958	−	0.953637i	$-0.402694\pi$
$409$	12.1730	0.601917	0.300958	+	0.953637i	$0.402694\pi$
$410$	4.76509	0.235331
$411$	−12.4437	−0.613801
$412$	−0.522900	−0.0257614
$413$	12.6676	0.623331
$414$	−0.353456	−0.0173714
$415$	9.10026	0.446714
$416$	−0.300372	−0.0147269
$417$	17.2188	0.843209
$418$	0	0
$419$	17.1854	0.839561	0.419781	−	0.907626i	$-0.362107\pi$
$419$	17.1854	0.839561	0.419781	+	0.907626i	$0.362107\pi$
$420$	−3.00000	−0.146385
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	−15.7775	−0.768035
$423$	−0.123644	−0.00601177
$424$	7.62178	0.370147
$425$	3.76509	0.182634
$426$	−4.37822	−0.212125
$427$	−10.0916	−0.488367
$428$	0.667585	0.0322689
$429$	0	0
$430$	−2.29528	−0.110688
$431$	32.7824	1.57907	0.789537	−	0.613703i	$-0.210321\pi$
$431$	32.7824	1.57907	0.789537	+	0.613703i	$0.210321\pi$
$432$	−5.28799	−0.254419
$433$	−15.0617	−0.723821	−0.361911	−	0.932213i	$-0.617875\pi$
$433$	−15.0617	−0.723821	−0.361911	+	0.932213i	$0.617875\pi$
$434$	16.7985	0.806354
$435$	16.1978	0.776624
$436$	9.53018	0.456413
$437$	−22.1978	−1.06186
$438$	−16.1075	−0.769648
$439$	−38.5526	−1.84002	−0.920008	−	0.391900i	$-0.871818\pi$
$439$	−38.5526	−1.84002	−0.920008	+	0.391900i	$0.871818\pi$
$440$	0	0
$441$	0.498141	0.0237210
$442$	0.300372	0.0142872
$443$	−9.49086	−0.450924	−0.225462	−	0.974252i	$-0.572389\pi$
$443$	−9.49086	−0.450924	−0.225462	+	0.974252i	$0.572389\pi$
$444$	−7.11993	−0.337897
$445$	1.76509	0.0836734
$446$	1.81089	0.0857482
$447$	−7.04442	−0.333190
$448$	1.58836	0.0750431
$449$	−12.0668	−0.569469	−0.284735	−	0.958606i	$-0.591906\pi$
$449$	−12.0668	−0.569469	−0.284735	+	0.958606i	$0.591906\pi$
$450$	−0.418920	−0.0197481
$451$	0	0
$452$	−13.6218	−0.640715
$453$	23.7527	1.11600
$454$	−3.24357	−0.152228
$455$	−0.530184	−0.0248554
$456$	−11.8764	−0.556161
$457$	30.8799	1.44450	0.722251	−	0.691631i	$-0.243108\pi$
$457$	30.8799	1.44450	0.722251	+	0.691631i	$0.243108\pi$
$458$	2.95420	0.138041
$459$	5.28799	0.246822
$460$	3.53018	0.164596
$461$	26.4486	1.23184	0.615918	−	0.787811i	$-0.288785\pi$
$461$	26.4486	1.23184	0.615918	+	0.787811i	$0.288785\pi$
$462$	0	0
$463$	38.8268	1.80444	0.902218	−	0.431280i	$-0.141938\pi$
$463$	38.8268	1.80444	0.902218	+	0.431280i	$0.141938\pi$
$464$	−8.57598	−0.398130
$465$	19.9752	0.926330
$466$	−9.47710	−0.439018
$467$	12.2473	0.566737	0.283368	−	0.959011i	$-0.408548\pi$
$467$	12.2473	0.566737	0.283368	+	0.959011i	$0.408548\pi$
$468$	−0.0334206	−0.00154487
$469$	19.9752	0.922371
$470$	1.23491	0.0569621
$471$	−28.1730	−1.29814
$472$	−7.97524	−0.367090
$473$	0	0
$474$	−4.28799	−0.196954
$475$	−26.3090	−1.20714
$476$	−1.58836	−0.0728025
$477$	0.848031	0.0388287
$478$	6.00000	0.274434
$479$	−0.614501	−0.0280773	−0.0140386	−	0.999901i	$-0.504469\pi$
$479$	−0.614501	−0.0280773	−0.0140386	+	0.999901i	$0.504469\pi$
$480$	1.88874	0.0862086
$481$	−1.25829	−0.0573731
$482$	7.90840	0.360218
$483$	−8.57598	−0.390221
$484$	0	0
$485$	−2.22253	−0.100920
$486$	−1.15569	−0.0524230
$487$	−0.197769	−0.00896176	−0.00448088	−	0.999990i	$-0.501426\pi$
$487$	−0.197769	−0.00896176	−0.00448088	+	0.999990i	$0.501426\pi$
$488$	6.35346	0.287607
$489$	23.6414	1.06910
$490$	−4.97524	−0.224758
$491$	7.53156	0.339895	0.169947	−	0.985453i	$-0.445640\pi$
$491$	7.53156	0.339895	0.169947	+	0.985453i	$0.445640\pi$
$492$	7.28799	0.328568
$493$	8.57598	0.386243
$494$	−2.09888	−0.0944333
$495$	0	0
$496$	−10.5760	−0.474876
$497$	4.09160	0.183533
$498$	13.9184	0.623700
$499$	−14.9011	−0.667066	−0.333533	−	0.942739i	$-0.608241\pi$
$499$	−14.9011	−0.667066	−0.333533	+	0.942739i	$0.608241\pi$
$500$	9.74033	0.435601
$501$	22.9986	1.02750
$502$	−18.1730	−0.811101
$503$	−27.6304	−1.23198	−0.615990	−	0.787754i	$-0.711244\pi$
$503$	−27.6304	−1.23198	−0.615990	+	0.787754i	$0.711244\pi$
$504$	0.176728	0.00787209
$505$	7.62769	0.339428
$506$	0	0
$507$	−21.9418	−0.974470
$508$	−3.84431	−0.170564
$509$	−24.5265	−1.08712	−0.543558	−	0.839371i	$-0.682923\pi$
$509$	−24.5265	−1.08712	−0.543558	+	0.839371i	$0.682923\pi$
$510$	−1.88874	−0.0836346
$511$	15.0531	0.665909
$512$	−1.00000	−0.0441942
$513$	−36.9505	−1.63140
$514$	29.4720	1.29995
$515$	0.581080	0.0256055
$516$	−3.51052	−0.154542
$517$	0	0
$518$	6.65383	0.292352
$519$	−25.9084	−1.13725
$520$	0.333792	0.0146378
$521$	38.0814	1.66838	0.834188	−	0.551480i	$-0.185937\pi$
$521$	38.0814	1.66838	0.834188	+	0.551480i	$0.185937\pi$
$522$	−0.954200	−0.0417642
$523$	−16.1075	−0.704334	−0.352167	−	0.935937i	$-0.614555\pi$
$523$	−16.1075	−0.704334	−0.352167	+	0.935937i	$0.614555\pi$
$524$	−19.7280	−0.861820
$525$	−10.1643	−0.443609
$526$	−8.57598	−0.373931
$527$	10.5760	0.460697
$528$	0	0
$529$	−12.9084	−0.561235
$530$	−8.46982	−0.367905
$531$	−0.887358	−0.0385081
$532$	11.0989	0.481198
$533$	1.28799	0.0557891
$534$	2.69963	0.116824
$535$	−0.741863	−0.0320735
$536$	−12.5760	−0.543200
$537$	−34.3708	−1.48321
$538$	−12.1396	−0.523375
$539$	0	0
$540$	5.87636	0.252878
$541$	36.8799	1.58559	0.792796	−	0.609487i	$-0.208625\pi$
$541$	36.8799	1.58559	0.792796	+	0.609487i	$0.208625\pi$
$542$	14.5091	0.623221
$543$	−18.7083	−0.802850
$544$	1.00000	0.0428746
$545$	−10.5906	−0.453649
$546$	−0.810892	−0.0347030
$547$	5.08513	0.217424	0.108712	−	0.994073i	$-0.465327\pi$
$547$	5.08513	0.217424	0.108712	+	0.994073i	$0.465327\pi$
$548$	−7.32141	−0.312755
$549$	0.706912	0.0301703
$550$	0	0
$551$	−59.9257	−2.55292
$552$	5.39926	0.229808
$553$	4.00728	0.170407
$554$	29.5723	1.25641
$555$	7.91212	0.335851
$556$	10.1309	0.429647
$557$	12.1964	0.516778	0.258389	−	0.966041i	$-0.416808\pi$
$557$	12.1964	0.516778	0.258389	+	0.966041i	$0.416808\pi$
$558$	−1.17673	−0.0498149
$559$	−0.620407	−0.0262404
$560$	−1.76509	−0.0745887
$561$	0	0
$562$	31.4413	1.32627
$563$	−22.9556	−0.967462	−0.483731	−	0.875217i	$-0.660719\pi$
$563$	−22.9556	−0.967462	−0.483731	+	0.875217i	$0.660719\pi$
$564$	1.88874	0.0795301
$565$	15.1374	0.636835
$566$	−15.1767	−0.637925
$567$	−13.7454	−0.577254
$568$	−2.57598	−0.108086
$569$	−36.0000	−1.50920	−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000	−1.50920	−0.754599	+	0.656186i	$0.772169\pi$
$570$	13.1978	0.552794
$571$	−9.17673	−0.384034	−0.192017	−	0.981392i	$-0.561503\pi$
$571$	−9.17673	−0.384034	−0.192017	+	0.981392i	$0.561503\pi$
$572$	0	0
$573$	−35.0617	−1.46473
$574$	−6.81089	−0.284281
$575$	11.9607	0.498795
$576$	−0.111264	−0.00463601
$577$	−38.3126	−1.59497	−0.797487	−	0.603336i	$-0.793838\pi$
$577$	−38.3126	−1.59497	−0.797487	+	0.603336i	$0.793838\pi$
$578$	−1.00000	−0.0415945
$579$	5.30903	0.220636
$580$	9.53018	0.395719
$581$	−13.0073	−0.539633
$582$	−3.39926	−0.140904
$583$	0	0
$584$	−9.47710	−0.392165
$585$	0.0371391	0.00153551
$586$	3.62041	0.149558
$587$	1.62178	0.0669382	0.0334691	−	0.999440i	$-0.489344\pi$
$587$	1.62178	0.0669382	0.0334691	+	0.999440i	$0.489344\pi$
$588$	−7.60940	−0.313807
$589$	−73.9010	−3.04504
$590$	8.86260	0.364867
$591$	−14.5760	−0.599576
$592$	−4.18911	−0.172171
$593$	23.5329	0.966382	0.483191	−	0.875515i	$-0.339478\pi$
$593$	23.5329	0.966382	0.483191	+	0.875515i	$0.339478\pi$
$594$	0	0
$595$	1.76509	0.0723617
$596$	−4.14468	−0.169773
$597$	13.9332	0.570246
$598$	0.954200	0.0390201
$599$	13.9084	0.568282	0.284141	−	0.958783i	$-0.408292\pi$
$599$	13.9084	0.568282	0.284141	+	0.958783i	$0.408292\pi$
$600$	6.39926	0.261249
$601$	−4.23119	−0.172594	−0.0862969	−	0.996269i	$-0.527503\pi$
$601$	−4.23119	−0.172594	−0.0862969	+	0.996269i	$0.527503\pi$
$602$	3.28071	0.133712
$603$	−1.39926	−0.0569821
$604$	13.9752	0.568644
$605$	0	0
$606$	11.6662	0.473907
$607$	35.1286	1.42583	0.712913	−	0.701253i	$-0.247376\pi$
$607$	35.1286	1.42583	0.712913	+	0.701253i	$0.247376\pi$
$608$	−6.98762	−0.283385
$609$	−23.1520	−0.938165
$610$	−7.06037	−0.285866
$611$	0.333792	0.0135038
$612$	0.111264	0.00449759
$613$	−42.6661	−1.72327	−0.861633	−	0.507532i	$-0.830558\pi$
$613$	−42.6661	−1.72327	−0.861633	+	0.507532i	$0.830558\pi$
$614$	33.2967	1.34374
$615$	−8.09888	−0.326579
$616$	0	0
$617$	−16.0248	−0.645133	−0.322566	−	0.946547i	$-0.604546\pi$
$617$	−16.0248	−0.645133	−0.322566	+	0.946547i	$0.604546\pi$
$618$	0.888736	0.0357502
$619$	14.9011	0.598926	0.299463	−	0.954108i	$-0.403192\pi$
$619$	14.9011	0.598926	0.299463	+	0.954108i	$0.403192\pi$
$620$	11.7527	0.472000
$621$	16.7985	0.674101
$622$	7.86907	0.315521
$623$	−2.52290	−0.101078
$624$	0.510520	0.0204372
$625$	8.00138	0.320055
$626$	−23.4413	−0.936904
$627$	0	0
$628$	−16.5760	−0.661454
$629$	4.18911	0.167031
$630$	−0.196391	−0.00782442
$631$	27.9788	1.11382	0.556909	−	0.830573i	$-0.311987\pi$
$631$	27.9788	1.11382	0.556909	+	0.830573i	$0.311987\pi$
$632$	−2.52290	−0.100356
$633$	26.8158	1.06583
$634$	2.89011	0.114781
$635$	4.27205	0.169531
$636$	−12.9542	−0.513667
$637$	−1.34479	−0.0532827
$638$	0	0
$639$	−0.286615	−0.0113383
$640$	1.11126	0.0439266
$641$	−17.1520	−0.677462	−0.338731	−	0.940883i	$-0.609998\pi$
$641$	−17.1520	−0.677462	−0.338731	+	0.940883i	$0.609998\pi$
$642$	−1.13465	−0.0447809
$643$	−9.15197	−0.360918	−0.180459	−	0.983582i	$-0.557758\pi$
$643$	−9.15197	−0.360918	−0.180459	+	0.983582i	$0.557758\pi$
$644$	−5.04580	−0.198832
$645$	3.90112	0.153606
$646$	6.98762	0.274924
$647$	−8.02832	−0.315626	−0.157813	−	0.987469i	$-0.550444\pi$
$647$	−8.02832	−0.315626	−0.157813	+	0.987469i	$0.550444\pi$
$648$	8.65383	0.339954
$649$	0	0
$650$	1.13093	0.0443586
$651$	−28.5512	−1.11901
$652$	13.9098	0.544749
$653$	2.67625	0.104730	0.0523648	−	0.998628i	$-0.483324\pi$
$653$	2.67625	0.104730	0.0523648	+	0.998628i	$0.483324\pi$
$654$	−16.1978	−0.633383
$655$	21.9230	0.856601
$656$	4.28799	0.167418
$657$	−1.05446	−0.0411385
$658$	−1.76509	−0.0688104
$659$	−30.4930	−1.18784	−0.593920	−	0.804524i	$-0.702421\pi$
$659$	−30.4930	−1.18784	−0.593920	+	0.804524i	$0.702421\pi$
$660$	0	0
$661$	−14.4871	−0.563484	−0.281742	−	0.959490i	$-0.590912\pi$
$661$	−14.4871	−0.563484	−0.281742	+	0.959490i	$0.590912\pi$
$662$	−10.2894	−0.399908
$663$	−0.510520	−0.0198270
$664$	8.18911	0.317799
$665$	−12.3338	−0.478284
$666$	−0.466098	−0.0180609
$667$	27.2436	1.05488
$668$	13.5316	0.523552
$669$	−3.07784	−0.118996
$670$	13.9752	0.539911
$671$	0	0
$672$	−2.69963	−0.104140
$673$	35.6625	1.37469	0.687344	−	0.726332i	$-0.258777\pi$
$673$	35.6625	1.37469	0.687344	+	0.726332i	$0.258777\pi$
$674$	−19.7120	−0.759278
$675$	19.9098	0.766328
$676$	−12.9098	−0.496530
$677$	−25.7280	−0.988806	−0.494403	−	0.869233i	$-0.664613\pi$
$677$	−25.7280	−0.988806	−0.494403	+	0.869233i	$0.664613\pi$
$678$	23.1520	0.889146
$679$	3.17673	0.121912
$680$	−1.11126	−0.0426150
$681$	5.51286	0.211253
$682$	0	0
$683$	−49.2843	−1.88581	−0.942905	−	0.333061i	$-0.891918\pi$
$683$	−49.2843	−1.88581	−0.942905	+	0.333061i	$0.891918\pi$
$684$	−0.777472	−0.0297274
$685$	8.13602	0.310861
$686$	18.2298	0.696017
$687$	−5.02104	−0.191565
$688$	−2.06546	−0.0787451
$689$	−2.28937	−0.0872180
$690$	−6.00000	−0.228416
$691$	−40.3126	−1.53356	−0.766782	−	0.641908i	$-0.778143\pi$
$691$	−40.3126	−1.53356	−0.766782	+	0.641908i	$0.778143\pi$
$692$	−15.2436	−0.579474
$693$	0	0
$694$	−29.8196	−1.13193
$695$	−11.2581	−0.427045
$696$	14.5760	0.552501
$697$	−4.28799	−0.162419
$698$	33.2436	1.25829
$699$	16.1075	0.609243
$700$	−5.98034	−0.226035
$701$	10.7651	0.406592	0.203296	−	0.979117i	$-0.434835\pi$
$701$	10.7651	0.406592	0.203296	+	0.979117i	$0.434835\pi$
$702$	1.58836	0.0599489
$703$	−29.2719	−1.10401
$704$	0	0
$705$	−2.09888	−0.0790486
$706$	2.66621	0.100344
$707$	−10.9025	−0.410030
$708$	13.5549	0.509426
$709$	−20.2064	−0.758868	−0.379434	−	0.925219i	$-0.623881\pi$
$709$	−20.2064	−0.758868	−0.379434	+	0.925219i	$0.623881\pi$
$710$	2.86260	0.107431
$711$	−0.280708	−0.0105274
$712$	1.58836	0.0595265
$713$	33.5970	1.25822
$714$	2.69963	0.101031
$715$	0	0
$716$	−20.2225	−0.755751
$717$	−10.1978	−0.380843
$718$	−16.8654	−0.629409
$719$	24.1062	0.899008	0.449504	−	0.893278i	$-0.351601\pi$
$719$	24.1062	0.899008	0.449504	+	0.893278i	$0.351601\pi$
$720$	0.123644	0.00460793
$721$	−0.830556	−0.0309315
$722$	−29.8268	−1.11004
$723$	−13.4413	−0.499889
$724$	−11.0073	−0.409082
$725$	32.2894	1.19920
$726$	0	0
$727$	9.38550	0.348089	0.174044	−	0.984738i	$-0.444316\pi$
$727$	9.38550	0.348089	0.174044	+	0.984738i	$0.444316\pi$
$728$	−0.477100	−0.0176825
$729$	27.9257	1.03429
$730$	10.5316	0.389791
$731$	2.06546	0.0763939
$732$	−10.7985	−0.399125
$733$	25.2174	0.931427	0.465714	−	0.884936i	$-0.345798\pi$
$733$	25.2174	0.931427	0.465714	+	0.884936i	$0.345798\pi$
$734$	−14.9542	−0.551970
$735$	8.45606	0.311906
$736$	3.17673	0.117096
$737$	0	0
$738$	0.477100	0.0175623
$739$	−10.3920	−0.382275	−0.191137	−	0.981563i	$-0.561218\pi$
$739$	−10.3920	−0.382275	−0.191137	+	0.981563i	$0.561218\pi$
$740$	4.65521	0.171129
$741$	3.56732	0.131049
$742$	12.1062	0.444431
$743$	−6.86398	−0.251815	−0.125907	−	0.992042i	$-0.540184\pi$
$743$	−6.86398	−0.251815	−0.125907	+	0.992042i	$0.540184\pi$
$744$	17.9752	0.659004
$745$	4.60584	0.168745
$746$	7.52152	0.275383
$747$	0.911154	0.0333374
$748$	0	0
$749$	1.06037	0.0387450
$750$	−16.5549	−0.604501
$751$	34.1062	1.24455	0.622276	−	0.782798i	$-0.286208\pi$
$751$	34.1062	1.24455	0.622276	+	0.782798i	$0.286208\pi$
$752$	1.11126	0.0405236
$753$	30.8874	1.12560
$754$	2.57598	0.0938118
$755$	−15.5302	−0.565201
$756$	−8.39926	−0.305478
$757$	−37.2581	−1.35417	−0.677085	−	0.735905i	$-0.736757\pi$
$757$	−37.2581	−1.35417	−0.677085	+	0.735905i	$0.736757\pi$
$758$	8.57461	0.311444
$759$	0	0
$760$	7.76509	0.281669
$761$	15.0631	0.546038	0.273019	−	0.962009i	$-0.411978\pi$
$761$	15.0631	0.546038	0.273019	+	0.962009i	$0.411978\pi$
$762$	6.53390	0.236698
$763$	15.1374	0.548011
$764$	−20.6291	−0.746334
$765$	−0.123644	−0.00447035
$766$	−32.2619	−1.16567
$767$	2.39554	0.0864979
$768$	1.69963	0.0613300
$769$	15.3497	0.553526	0.276763	−	0.960938i	$-0.410738\pi$
$769$	15.3497	0.553526	0.276763	+	0.960938i	$0.410738\pi$
$770$	0	0
$771$	−50.0914	−1.80400
$772$	3.12364	0.112422
$773$	−48.1455	−1.73167	−0.865837	−	0.500327i	$-0.833213\pi$
$773$	−48.1455	−1.73167	−0.865837	+	0.500327i	$0.833213\pi$
$774$	−0.229812	−0.00826043
$775$	39.8196	1.43036
$776$	−2.00000	−0.0717958
$777$	−11.3090	−0.405709
$778$	−17.5054	−0.627600
$779$	29.9629	1.07353
$780$	−0.567323	−0.0203134
$781$	0	0
$782$	−3.17673	−0.113600
$783$	45.3497	1.62067
$784$	−4.47710	−0.159896
$785$	18.4203	0.657449
$786$	33.5302	1.19598
$787$	43.5082	1.55090	0.775450	−	0.631410i	$-0.217523\pi$
$787$	43.5082	1.55090	0.775450	+	0.631410i	$0.217523\pi$
$788$	−8.57598	−0.305507
$789$	14.5760	0.518919
$790$	2.80361	0.0997479
$791$	−21.6364	−0.769300
$792$	0	0
$793$	−1.90840	−0.0677692
$794$	7.81089	0.277198
$795$	14.3955	0.510557
$796$	8.19777	0.290562
$797$	11.3324	0.401415	0.200707	−	0.979651i	$-0.435676\pi$
$797$	11.3324	0.401415	0.200707	+	0.979651i	$0.435676\pi$
$798$	−18.8640	−0.667778
$799$	−1.11126	−0.0393137
$800$	3.76509	0.133116
$801$	0.176728	0.00624438
$802$	9.10989	0.321681
$803$	0	0
$804$	21.3745	0.753820
$805$	5.60722	0.197628
$806$	3.17673	0.111895
$807$	20.6328	0.726308
$808$	6.86398	0.241474
$809$	19.9084	0.699942	0.349971	−	0.936761i	$-0.386191\pi$
$809$	19.9084	0.699942	0.349971	+	0.936761i	$0.386191\pi$
$810$	−9.61669	−0.337896
$811$	52.2371	1.83429	0.917146	−	0.398551i	$-0.130487\pi$
$811$	52.2371	1.83429	0.917146	+	0.398551i	$0.130487\pi$
$812$	−13.6218	−0.478031
$813$	−24.6601	−0.864869
$814$	0	0
$815$	−15.4574	−0.541450
$816$	−1.69963	−0.0594989
$817$	−14.4327	−0.504935
$818$	−12.1730	−0.425619
$819$	−0.0530841	−0.00185491
$820$	−4.76509	−0.166404
$821$	28.8654	1.00741	0.503704	−	0.863876i	$-0.331970\pi$
$821$	28.8654	1.00741	0.503704	+	0.863876i	$0.331970\pi$
$822$	12.4437	0.434023
$823$	29.6364	1.03306	0.516529	−	0.856270i	$-0.327224\pi$
$823$	29.6364	1.03306	0.516529	+	0.856270i	$0.327224\pi$
$824$	0.522900	0.0182161
$825$	0	0
$826$	−12.6676	−0.440762
$827$	36.3928	1.26550	0.632751	−	0.774356i	$-0.281926\pi$
$827$	36.3928	1.26550	0.632751	+	0.774356i	$0.281926\pi$
$828$	0.353456	0.0122834
$829$	33.9084	1.17769	0.588844	−	0.808247i	$-0.299583\pi$
$829$	33.9084	1.17769	0.588844	+	0.808247i	$0.299583\pi$
$830$	−9.10026	−0.315875
$831$	−50.2619	−1.74356
$832$	0.300372	0.0104135
$833$	4.47710	0.155122
$834$	−17.2188	−0.596239
$835$	−15.0371	−0.520382
$836$	0	0
$837$	55.9257	1.93308
$838$	−17.1854	−0.593659
$839$	−44.8158	−1.54721	−0.773607	−	0.633665i	$-0.781550\pi$
$839$	−44.8158	−1.54721	−0.773607	+	0.633665i	$0.781550\pi$
$840$	3.00000	0.103510
$841$	44.5475	1.53612
$842$	−10.0000	−0.344623
$843$	−53.4386	−1.84052
$844$	15.7775	0.543083
$845$	14.3462	0.493523
$846$	0.123644	0.00425096
$847$	0	0
$848$	−7.62178	−0.261733
$849$	25.7948	0.885275
$850$	−3.76509	−0.129142
$851$	13.3077	0.456181
$852$	4.37822	0.149995
$853$	3.00372	0.102845	0.0514227	−	0.998677i	$-0.483624\pi$
$853$	3.00372	0.102845	0.0514227	+	0.998677i	$0.483624\pi$
$854$	10.0916	0.345327
$855$	0.863976	0.0295474
$856$	−0.667585	−0.0228176
$857$	45.4500	1.55254	0.776271	−	0.630399i	$-0.217109\pi$
$857$	45.4500	1.55254	0.776271	+	0.630399i	$0.217109\pi$
$858$	0	0
$859$	10.8626	0.370627	0.185314	−	0.982679i	$-0.440670\pi$
$859$	10.8626	0.370627	0.185314	+	0.982679i	$0.440670\pi$
$860$	2.29528	0.0782683
$861$	11.5760	0.394508
$862$	−32.7824	−1.11657
$863$	19.4079	0.660653	0.330327	−	0.943867i	$-0.392841\pi$
$863$	19.4079	0.660653	0.330327	+	0.943867i	$0.392841\pi$
$864$	5.28799	0.179901
$865$	16.9396	0.575965
$866$	15.0617	0.511819
$867$	1.69963	0.0577224
$868$	−16.7985	−0.570179
$869$	0	0
$870$	−16.1978	−0.549156
$871$	3.77747	0.127995
$872$	−9.53018	−0.322733
$873$	−0.222528	−0.00753144
$874$	22.1978	0.750851
$875$	15.4712	0.523022
$876$	16.1075	0.544224
$877$	39.1693	1.32265	0.661326	−	0.750098i	$-0.269994\pi$
$877$	39.1693	1.32265	0.661326	+	0.750098i	$0.269994\pi$
$878$	38.5526	1.30109
$879$	−6.15335	−0.207547
$880$	0	0
$881$	−49.0210	−1.65156	−0.825780	−	0.563992i	$-0.809265\pi$
$881$	−49.0210	−1.65156	−0.825780	+	0.563992i	$0.809265\pi$
$882$	−0.498141	−0.0167733
$883$	10.2866	0.346172	0.173086	−	0.984907i	$-0.444626\pi$
$883$	10.2866	0.346172	0.173086	+	0.984907i	$0.444626\pi$
$884$	−0.300372	−0.0101026
$885$	−15.0631	−0.506341
$886$	9.49086	0.318852
$887$	−40.7651	−1.36876	−0.684379	−	0.729127i	$-0.739927\pi$
$887$	−40.7651	−1.36876	−0.684379	+	0.729127i	$0.739927\pi$
$888$	7.11993	0.238929
$889$	−6.10617	−0.204794
$890$	−1.76509	−0.0591660
$891$	0	0
$892$	−1.81089	−0.0606332
$893$	7.76509	0.259849
$894$	7.04442	0.235601
$895$	22.4726	0.751175
$896$	−1.58836	−0.0530635
$897$	−1.62178	−0.0541498
$898$	12.0668	0.402676
$899$	90.6995	3.02500
$900$	0.418920	0.0139640
$901$	7.62178	0.253918
$902$	0	0
$903$	−5.57598	−0.185557
$904$	13.6218	0.453054
$905$	12.2320	0.406605
$906$	−23.7527	−0.789131
$907$	4.80951	0.159697	0.0798487	−	0.996807i	$-0.474556\pi$
$907$	4.80951	0.159697	0.0798487	+	0.996807i	$0.474556\pi$
$908$	3.24357	0.107642
$909$	0.763715	0.0253308
$910$	0.530184	0.0175754
$911$	29.2581	0.969365	0.484683	−	0.874690i	$-0.338935\pi$
$911$	29.2581	0.969365	0.484683	+	0.874690i	$0.338935\pi$
$912$	11.8764	0.393265
$913$	0	0
$914$	−30.8799	−1.02142
$915$	12.0000	0.396708
$916$	−2.95420	−0.0976095
$917$	−31.3352	−1.03478
$918$	−5.28799	−0.174530
$919$	37.0604	1.22251	0.611254	−	0.791434i	$-0.290665\pi$
$919$	37.0604	1.22251	0.611254	+	0.791434i	$0.290665\pi$
$920$	−3.53018	−0.116387
$921$	−56.5919	−1.86477
$922$	−26.4486	−0.871039
$923$	0.773753	0.0254684
$924$	0	0
$925$	15.7724	0.518593
$926$	−38.8268	−1.27593
$927$	0.0581800	0.00191088
$928$	8.57598	0.281520
$929$	−7.06037	−0.231643	−0.115822	−	0.993270i	$-0.536950\pi$
$929$	−7.06037	−0.231643	−0.115822	+	0.993270i	$0.536950\pi$
$930$	−19.9752	−0.655014
$931$	−31.2843	−1.02530
$932$	9.47710	0.310433
$933$	−13.3745	−0.437861
$934$	−12.2473	−0.400743
$935$	0	0
$936$	0.0334206	0.00109239
$937$	−48.8131	−1.59465	−0.797327	−	0.603548i	$-0.793753\pi$
$937$	−48.8131	−1.59465	−0.797327	+	0.603548i	$0.793753\pi$
$938$	−19.9752	−0.652215
$939$	39.8416	1.30018
$940$	−1.23491	−0.0402783
$941$	−32.5760	−1.06195	−0.530973	−	0.847389i	$-0.678174\pi$
$941$	−32.5760	−1.06195	−0.530973	+	0.847389i	$0.678174\pi$
$942$	28.1730	0.917926
$943$	−13.6218	−0.443586
$944$	7.97524	0.259572
$945$	9.33379	0.303628
$946$	0	0
$947$	−19.9986	−0.649868	−0.324934	−	0.945737i	$-0.605342\pi$
$947$	−19.9986	−0.649868	−0.324934	+	0.945737i	$0.605342\pi$
$948$	4.28799	0.139268
$949$	2.84665	0.0924063
$950$	26.3090	0.853578
$951$	−4.91212	−0.159286
$952$	1.58836	0.0514792
$953$	−17.1520	−0.555607	−0.277803	−	0.960638i	$-0.589606\pi$
$953$	−17.1520	−0.555607	−0.277803	+	0.960638i	$0.589606\pi$
$954$	−0.848031	−0.0274560
$955$	22.9243	0.741814
$956$	−6.00000	−0.194054
$957$	0	0
$958$	0.614501	0.0198536
$959$	−11.6291	−0.375522
$960$	−1.88874	−0.0609587
$961$	80.8514	2.60811
$962$	1.25829	0.0405689
$963$	−0.0742783	−0.00239358
$964$	−7.90840	−0.254712
$965$	−3.47119	−0.111742
$966$	8.57598	0.275928
$967$	51.2755	1.64891	0.824454	−	0.565929i	$-0.191482\pi$
$967$	51.2755	1.64891	0.824454	+	0.565929i	$0.191482\pi$
$968$	0	0
$969$	−11.8764	−0.381524
$970$	2.22253	0.0713611
$971$	27.9112	0.895712	0.447856	−	0.894106i	$-0.352188\pi$
$971$	27.9112	0.895712	0.447856	+	0.894106i	$0.352188\pi$
$972$	1.15569	0.0370687
$973$	16.0916	0.515873
$974$	0.197769	0.00633692
$975$	−1.92216	−0.0615583
$976$	−6.35346	−0.203369
$977$	−14.4858	−0.463441	−0.231720	−	0.972782i	$-0.574435\pi$
$977$	−14.4858	−0.463441	−0.231720	+	0.972782i	$0.574435\pi$
$978$	−23.6414	−0.755970
$979$	0	0
$980$	4.97524	0.158928
$981$	−1.06037	−0.0338549
$982$	−7.53156	−0.240342
$983$	−20.5760	−0.656272	−0.328136	−	0.944630i	$-0.606420\pi$
$983$	−20.5760	−0.656272	−0.328136	+	0.944630i	$0.606420\pi$
$984$	−7.28799	−0.232333
$985$	9.53018	0.303657
$986$	−8.57598	−0.273115
$987$	3.00000	0.0954911
$988$	2.09888	0.0667744
$989$	6.56142	0.208641
$990$	0	0
$991$	−49.9257	−1.58594	−0.792971	−	0.609259i	$-0.791467\pi$
$991$	−49.9257	−1.58594	−0.792971	+	0.609259i	$0.791467\pi$
$992$	10.5760	0.335788
$993$	17.4881	0.554968
$994$	−4.09160	−0.129778
$995$	−9.10989	−0.288803
$996$	−13.9184	−0.441023
$997$	54.5265	1.72687	0.863435	−	0.504460i	$-0.168308\pi$
$997$	54.5265	1.72687	0.863435	+	0.504460i	$0.168308\pi$
$998$	14.9011	0.471687
$999$	22.1520	0.700857

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4114.2.a.o.1.3	✓	3
11.10	odd	2		4114.2.a.r.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4114.2.a.o.1.3	✓	3	1.1	even	1	trivial
4114.2.a.r.1.3	yes	3	11.10	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4114 = 2 \cdot 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4114.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.69963 q^{3} +1.00000 q^{4} -1.11126 q^{5} -1.69963 q^{6} +1.58836 q^{7} -1.00000 q^{8} -0.111264 q^{9} +1.11126 q^{10} +1.69963 q^{12} +0.300372 q^{13} -1.58836 q^{14} -1.88874 q^{15} +1.00000 q^{16} -1.00000 q^{17} +0.111264 q^{18} +6.98762 q^{19} -1.11126 q^{20} +2.69963 q^{21} -3.17673 q^{23} -1.69963 q^{24} -3.76509 q^{25} -0.300372 q^{26} -5.28799 q^{27} +1.58836 q^{28} -8.57598 q^{29} +1.88874 q^{30} -10.5760 q^{31} -1.00000 q^{32} +1.00000 q^{34} -1.76509 q^{35} -0.111264 q^{36} -4.18911 q^{37} -6.98762 q^{38} +0.510520 q^{39} +1.11126 q^{40} +4.28799 q^{41} -2.69963 q^{42} -2.06546 q^{43} +0.123644 q^{45} +3.17673 q^{46} +1.11126 q^{47} +1.69963 q^{48} -4.47710 q^{49} +3.76509 q^{50} -1.69963 q^{51} +0.300372 q^{52} -7.62178 q^{53} +5.28799 q^{54} -1.58836 q^{56} +11.8764 q^{57} +8.57598 q^{58} +7.97524 q^{59} -1.88874 q^{60} -6.35346 q^{61} +10.5760 q^{62} -0.176728 q^{63} +1.00000 q^{64} -0.333792 q^{65} +12.5760 q^{67} -1.00000 q^{68} -5.39926 q^{69} +1.76509 q^{70} +2.57598 q^{71} +0.111264 q^{72} +9.47710 q^{73} +4.18911 q^{74} -6.39926 q^{75} +6.98762 q^{76} -0.510520 q^{78} +2.52290 q^{79} -1.11126 q^{80} -8.65383 q^{81} -4.28799 q^{82} -8.18911 q^{83} +2.69963 q^{84} +1.11126 q^{85} +2.06546 q^{86} -14.5760 q^{87} -1.58836 q^{89} -0.123644 q^{90} +0.477100 q^{91} -3.17673 q^{92} -17.9752 q^{93} -1.11126 q^{94} -7.76509 q^{95} -1.69963 q^{96} +2.00000 q^{97} +4.47710 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - q^{3} + 3 q^{4} - 3 q^{5} + q^{6} - q^{7} - 3 q^{8} + 3 q^{10} - q^{12} + 7 q^{13} + q^{14} - 6 q^{15} + 3 q^{16} - 3 q^{17} + 3 q^{19} - 3 q^{20} + 2 q^{21} + 2 q^{23} + q^{24} + 6 q^{25}+ \cdots + 8 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 - q^3 + 3 * q^4 - 3 * q^5 + q^6 - q^7 - 3 * q^8 + 3 * q^10 - q^12 + 7 * q^13 + q^14 - 6 * q^15 + 3 * q^16 - 3 * q^17 + 3 * q^19 - 3 * q^20 + 2 * q^21 + 2 * q^23 + q^24 + 6 * q^25 - 7 * q^26 - 4 * q^27 - q^28 - 2 * q^29 + 6 * q^30 - 8 * q^31 - 3 * q^32 + 3 * q^34 + 12 * q^35 - 19 * q^37 - 3 * q^38 - 11 * q^39 + 3 * q^40 + q^41 - 2 * q^42 + 5 * q^43 + 18 * q^45 - 2 * q^46 + 3 * q^47 - q^48 - 8 * q^49 - 6 * q^50 + q^51 + 7 * q^52 - 10 * q^53 + 4 * q^54 + q^56 + 18 * q^57 + 2 * q^58 - 12 * q^59 - 6 * q^60 + 4 * q^61 + 8 * q^62 + 11 * q^63 + 3 * q^64 + 14 * q^67 - 3 * q^68 - 4 * q^69 - 12 * q^70 - 16 * q^71 + 23 * q^73 + 19 * q^74 - 7 * q^75 + 3 * q^76 + 11 * q^78 + 13 * q^79 - 3 * q^80 - 9 * q^81 - q^82 - 31 * q^83 + 2 * q^84 + 3 * q^85 - 5 * q^86 - 20 * q^87 + q^89 - 18 * q^90 - 4 * q^91 + 2 * q^92 - 18 * q^93 - 3 * q^94 - 6 * q^95 + q^96 + 6 * q^97 + 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more