LMFDB - Embedded newform 2299.2.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2299,2,Mod(1,2299)] 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("2299.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2299, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2299 = 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2299.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,6,-1,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.3576074247$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	2299.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.23607 q^{2} -1.23607 q^{3} +3.00000 q^{4} +0.618034 q^{5} -2.76393 q^{6} -2.61803 q^{7} +2.23607 q^{8} -1.47214 q^{9} +1.38197 q^{10} -3.70820 q^{12} +6.47214 q^{13} -5.85410 q^{14} -0.763932 q^{15} -1.00000 q^{16} +8.09017 q^{17} -3.29180 q^{18} +1.00000 q^{19} +1.85410 q^{20} +3.23607 q^{21} +1.61803 q^{23} -2.76393 q^{24} -4.61803 q^{25} +14.4721 q^{26} +5.52786 q^{27} -7.85410 q^{28} -0.472136 q^{29} -1.70820 q^{30} +8.00000 q^{31} -6.70820 q^{32} +18.0902 q^{34} -1.61803 q^{35} -4.41641 q^{36} -2.00000 q^{37} +2.23607 q^{38} -8.00000 q^{39} +1.38197 q^{40} -1.70820 q^{41} +7.23607 q^{42} +8.09017 q^{43} -0.909830 q^{45} +3.61803 q^{46} +6.38197 q^{47} +1.23607 q^{48} -0.145898 q^{49} -10.3262 q^{50} -10.0000 q^{51} +19.4164 q^{52} +10.4721 q^{53} +12.3607 q^{54} -5.85410 q^{56} -1.23607 q^{57} -1.05573 q^{58} +3.52786 q^{59} -2.29180 q^{60} +6.38197 q^{61} +17.8885 q^{62} +3.85410 q^{63} -13.0000 q^{64} +4.00000 q^{65} +0.472136 q^{67} +24.2705 q^{68} -2.00000 q^{69} -3.61803 q^{70} -1.70820 q^{71} -3.29180 q^{72} +12.4721 q^{73} -4.47214 q^{74} +5.70820 q^{75} +3.00000 q^{76} -17.8885 q^{78} -14.9443 q^{79} -0.618034 q^{80} -2.41641 q^{81} -3.81966 q^{82} -10.3262 q^{83} +9.70820 q^{84} +5.00000 q^{85} +18.0902 q^{86} +0.583592 q^{87} +1.23607 q^{89} -2.03444 q^{90} -16.9443 q^{91} +4.85410 q^{92} -9.88854 q^{93} +14.2705 q^{94} +0.618034 q^{95} +8.29180 q^{96} +10.1803 q^{97} -0.326238 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 6 q^{4} - q^{5} - 10 q^{6} - 3 q^{7} + 6 q^{9} + 5 q^{10} + 6 q^{12} + 4 q^{13} - 5 q^{14} - 6 q^{15} - 2 q^{16} + 5 q^{17} - 20 q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{21} + q^{23} - 10 q^{24}+ \cdots + 15 q^{98}+O(q^{100}) $ 2 * q + 2 * q^3 + 6 * q^4 - q^5 - 10 * q^6 - 3 * q^7 + 6 * q^9 + 5 * q^10 + 6 * q^12 + 4 * q^13 - 5 * q^14 - 6 * q^15 - 2 * q^16 + 5 * q^17 - 20 * q^18 + 2 * q^19 - 3 * q^20 + 2 * q^21 + q^23 - 10 * q^24 - 7 * q^25 + 20 * q^26 + 20 * q^27 - 9 * q^28 + 8 * q^29 + 10 * q^30 + 16 * q^31 + 25 * q^34 - q^35 + 18 * q^36 - 4 * q^37 - 16 * q^39 + 5 * q^40 + 10 * q^41 + 10 * q^42 + 5 * q^43 - 13 * q^45 + 5 * q^46 + 15 * q^47 - 2 * q^48 - 7 * q^49 - 5 * q^50 - 20 * q^51 + 12 * q^52 + 12 * q^53 - 20 * q^54 - 5 * q^56 + 2 * q^57 - 20 * q^58 + 16 * q^59 - 18 * q^60 + 15 * q^61 + q^63 - 26 * q^64 + 8 * q^65 - 8 * q^67 + 15 * q^68 - 4 * q^69 - 5 * q^70 + 10 * q^71 - 20 * q^72 + 16 * q^73 - 2 * q^75 + 6 * q^76 - 12 * q^79 + q^80 + 22 * q^81 - 30 * q^82 - 5 * q^83 + 6 * q^84 + 10 * q^85 + 25 * q^86 + 28 * q^87 - 2 * q^89 + 25 * q^90 - 16 * q^91 + 3 * q^92 + 16 * q^93 - 5 * q^94 - q^95 + 30 * q^96 - 2 * q^97 + 15 * 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 $

$2$	2.23607	1.58114	0.790569	−	0.612372i	$-0.209785\pi$
$2$	2.23607	1.58114	0.790569	+	0.612372i	$0.209785\pi$
$3$	−1.23607	−0.713644	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	−1.23607	−0.713644	−0.356822	+	0.934172i	$0.616140\pi$
$4$	3.00000	1.50000
$5$	0.618034	0.276393	0.138197	−	0.990405i	$-0.455869\pi$
$5$	0.618034	0.276393	0.138197	+	0.990405i	$0.455869\pi$
$6$	−2.76393	−1.12837
$7$	−2.61803	−0.989524	−0.494762	−	0.869029i	$-0.664745\pi$
$7$	−2.61803	−0.989524	−0.494762	+	0.869029i	$0.664745\pi$
$8$	2.23607	0.790569
$9$	−1.47214	−0.490712
$10$	1.38197	0.437016
$11$	0	0
$11$	0	0
$12$	−3.70820	−1.07047
$13$	6.47214	1.79505	0.897524	−	0.440966i	$-0.145364\pi$
$13$	6.47214	1.79505	0.897524	+	0.440966i	$0.145364\pi$
$14$	−5.85410	−1.56457
$15$	−0.763932	−0.197246
$16$	−1.00000	−0.250000
$17$	8.09017	1.96215	0.981077	−	0.193617i	$-0.0620219\pi$
$17$	8.09017	1.96215	0.981077	+	0.193617i	$0.0620219\pi$
$18$	−3.29180	−0.775884
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.85410	0.414590
$21$	3.23607	0.706168
$22$	0	0
$23$	1.61803	0.337383	0.168692	−	0.985669i	$-0.446046\pi$
$23$	1.61803	0.337383	0.168692	+	0.985669i	$0.446046\pi$
$24$	−2.76393	−0.564185
$25$	−4.61803	−0.923607
$26$	14.4721	2.83822
$27$	5.52786	1.06384
$28$	−7.85410	−1.48429
$29$	−0.472136	−0.0876734	−0.0438367	−	0.999039i	$-0.513958\pi$
$29$	−0.472136	−0.0876734	−0.0438367	+	0.999039i	$0.513958\pi$
$30$	−1.70820	−0.311874
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−6.70820	−1.18585
$33$	0	0
$34$	18.0902	3.10244
$35$	−1.61803	−0.273498
$36$	−4.41641	−0.736068
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	2.23607	0.362738
$39$	−8.00000	−1.28103
$40$	1.38197	0.218508
$41$	−1.70820	−0.266777	−0.133388	−	0.991064i	$-0.542586\pi$
$41$	−1.70820	−0.266777	−0.133388	+	0.991064i	$0.542586\pi$
$42$	7.23607	1.11655
$43$	8.09017	1.23374	0.616870	−	0.787065i	$-0.288401\pi$
$43$	8.09017	1.23374	0.616870	+	0.787065i	$0.288401\pi$
$44$	0	0
$45$	−0.909830	−0.135629
$46$	3.61803	0.533450
$47$	6.38197	0.930905	0.465453	−	0.885073i	$-0.345892\pi$
$47$	6.38197	0.930905	0.465453	+	0.885073i	$0.345892\pi$
$48$	1.23607	0.178411
$49$	−0.145898	−0.0208426
$50$	−10.3262	−1.46035
$51$	−10.0000	−1.40028
$52$	19.4164	2.69257
$53$	10.4721	1.43846	0.719229	−	0.694773i	$-0.244495\pi$
$53$	10.4721	1.43846	0.719229	+	0.694773i	$0.244495\pi$
$54$	12.3607	1.68208
$55$	0	0
$56$	−5.85410	−0.782287
$57$	−1.23607	−0.163721
$58$	−1.05573	−0.138624
$59$	3.52786	0.459289	0.229644	−	0.973275i	$-0.426244\pi$
$59$	3.52786	0.459289	0.229644	+	0.973275i	$0.426244\pi$
$60$	−2.29180	−0.295870
$61$	6.38197	0.817127	0.408564	−	0.912730i	$-0.366030\pi$
$61$	6.38197	0.817127	0.408564	+	0.912730i	$0.366030\pi$
$62$	17.8885	2.27185
$63$	3.85410	0.485571
$64$	−13.0000	−1.62500
$65$	4.00000	0.496139
$66$	0	0
$67$	0.472136	0.0576806	0.0288403	−	0.999584i	$-0.490819\pi$
$67$	0.472136	0.0576806	0.0288403	+	0.999584i	$0.490819\pi$
$68$	24.2705	2.94323
$69$	−2.00000	−0.240772
$70$	−3.61803	−0.432438
$71$	−1.70820	−0.202727	−0.101363	−	0.994849i	$-0.532320\pi$
$71$	−1.70820	−0.202727	−0.101363	+	0.994849i	$0.532320\pi$
$72$	−3.29180	−0.387942
$73$	12.4721	1.45975	0.729877	−	0.683579i	$-0.239578\pi$
$73$	12.4721	1.45975	0.729877	+	0.683579i	$0.239578\pi$
$74$	−4.47214	−0.519875
$75$	5.70820	0.659127
$76$	3.00000	0.344124
$77$	0	0
$78$	−17.8885	−2.02548
$79$	−14.9443	−1.68136	−0.840681	−	0.541531i	$-0.817845\pi$
$79$	−14.9443	−1.68136	−0.840681	+	0.541531i	$0.817845\pi$
$80$	−0.618034	−0.0690983
$81$	−2.41641	−0.268490
$82$	−3.81966	−0.421811
$83$	−10.3262	−1.13345	−0.566726	−	0.823906i	$-0.691790\pi$
$83$	−10.3262	−1.13345	−0.566726	+	0.823906i	$0.691790\pi$
$84$	9.70820	1.05925
$85$	5.00000	0.542326
$86$	18.0902	1.95071
$87$	0.583592	0.0625676
$88$	0	0
$89$	1.23607	0.131023	0.0655115	−	0.997852i	$-0.479132\pi$
$89$	1.23607	0.131023	0.0655115	+	0.997852i	$0.479132\pi$
$90$	−2.03444	−0.214449
$91$	−16.9443	−1.77624
$92$	4.85410	0.506075
$93$	−9.88854	−1.02539
$94$	14.2705	1.47189
$95$	0.618034	0.0634089
$96$	8.29180	0.846278
$97$	10.1803	1.03366	0.516828	−	0.856089i	$-0.327113\pi$
$97$	10.1803	1.03366	0.516828	+	0.856089i	$0.327113\pi$
$98$	−0.326238	−0.0329550
$99$	0	0
$100$	−13.8541	−1.38541
$101$	−8.61803	−0.857526	−0.428763	−	0.903417i	$-0.641051\pi$
$101$	−8.61803	−0.857526	−0.428763	+	0.903417i	$0.641051\pi$
$102$	−22.3607	−2.21404
$103$	2.29180	0.225817	0.112909	−	0.993605i	$-0.463983\pi$
$103$	2.29180	0.225817	0.112909	+	0.993605i	$0.463983\pi$
$104$	14.4721	1.41911
$105$	2.00000	0.195180
$106$	23.4164	2.27440
$107$	−0.763932	−0.0738521	−0.0369260	−	0.999318i	$-0.511757\pi$
$107$	−0.763932	−0.0738521	−0.0369260	+	0.999318i	$0.511757\pi$
$108$	16.5836	1.59576
$109$	−6.94427	−0.665141	−0.332570	−	0.943078i	$-0.607916\pi$
$109$	−6.94427	−0.665141	−0.332570	+	0.943078i	$0.607916\pi$
$110$	0	0
$111$	2.47214	0.234645
$112$	2.61803	0.247381
$113$	−6.76393	−0.636297	−0.318149	−	0.948041i	$-0.603061\pi$
$113$	−6.76393	−0.636297	−0.318149	+	0.948041i	$0.603061\pi$
$114$	−2.76393	−0.258866
$115$	1.00000	0.0932505
$116$	−1.41641	−0.131510
$117$	−9.52786	−0.880851
$118$	7.88854	0.726199
$119$	−21.1803	−1.94160
$120$	−1.70820	−0.155937
$121$	0	0
$122$	14.2705	1.29199
$123$	2.11146	0.190384
$124$	24.0000	2.15526
$125$	−5.94427	−0.531672
$126$	8.61803	0.767755
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−15.6525	−1.38350
$129$	−10.0000	−0.880451
$130$	8.94427	0.784465
$131$	−18.0344	−1.57568	−0.787838	−	0.615882i	$-0.788800\pi$
$131$	−18.0344	−1.57568	−0.787838	+	0.615882i	$0.788800\pi$
$132$	0	0
$133$	−2.61803	−0.227012
$134$	1.05573	0.0912010
$135$	3.41641	0.294038
$136$	18.0902	1.55122
$137$	1.85410	0.158407	0.0792033	−	0.996858i	$-0.474762\pi$
$137$	1.85410	0.158407	0.0792033	+	0.996858i	$0.474762\pi$
$138$	−4.47214	−0.380693
$139$	−20.0902	−1.70403	−0.852013	−	0.523521i	$-0.824618\pi$
$139$	−20.0902	−1.70403	−0.852013	+	0.523521i	$0.824618\pi$
$140$	−4.85410	−0.410246
$141$	−7.88854	−0.664335
$142$	−3.81966	−0.320539
$143$	0	0
$144$	1.47214	0.122678
$145$	−0.291796	−0.0242323
$146$	27.8885	2.30807
$147$	0.180340	0.0148742
$148$	−6.00000	−0.493197
$149$	17.4164	1.42681	0.713404	−	0.700753i	$-0.247153\pi$
$149$	17.4164	1.42681	0.713404	+	0.700753i	$0.247153\pi$
$150$	12.7639	1.04217
$151$	2.76393	0.224926	0.112463	−	0.993656i	$-0.464126\pi$
$151$	2.76393	0.224926	0.112463	+	0.993656i	$0.464126\pi$
$152$	2.23607	0.181369
$153$	−11.9098	−0.962853
$154$	0	0
$155$	4.94427	0.397133
$156$	−24.0000	−1.92154
$157$	−5.56231	−0.443920	−0.221960	−	0.975056i	$-0.571245\pi$
$157$	−5.56231	−0.443920	−0.221960	+	0.975056i	$0.571245\pi$
$158$	−33.4164	−2.65847
$159$	−12.9443	−1.02655
$160$	−4.14590	−0.327762
$161$	−4.23607	−0.333849
$162$	−5.40325	−0.424520
$163$	6.32624	0.495509	0.247755	−	0.968823i	$-0.420307\pi$
$163$	6.32624	0.495509	0.247755	+	0.968823i	$0.420307\pi$
$164$	−5.12461	−0.400165
$165$	0	0
$166$	−23.0902	−1.79214
$167$	−1.81966	−0.140810	−0.0704048	−	0.997519i	$-0.522429\pi$
$167$	−1.81966	−0.140810	−0.0704048	+	0.997519i	$0.522429\pi$
$168$	7.23607	0.558275
$169$	28.8885	2.22220
$170$	11.1803	0.857493
$171$	−1.47214	−0.112577
$172$	24.2705	1.85061
$173$	18.4721	1.40441	0.702205	−	0.711975i	$-0.252199\pi$
$173$	18.4721	1.40441	0.702205	+	0.711975i	$0.252199\pi$
$174$	1.30495	0.0989281
$175$	12.0902	0.913931
$176$	0	0
$177$	−4.36068	−0.327769
$178$	2.76393	0.207165
$179$	16.1803	1.20938	0.604688	−	0.796463i	$-0.293298\pi$
$179$	16.1803	1.20938	0.604688	+	0.796463i	$0.293298\pi$
$180$	−2.72949	−0.203444
$181$	2.47214	0.183752	0.0918762	−	0.995770i	$-0.470714\pi$
$181$	2.47214	0.183752	0.0918762	+	0.995770i	$0.470714\pi$
$182$	−37.8885	−2.80849
$183$	−7.88854	−0.583138
$184$	3.61803	0.266725
$185$	−1.23607	−0.0908775
$186$	−22.1115	−1.62129
$187$	0	0
$188$	19.1459	1.39636
$189$	−14.4721	−1.05269
$190$	1.38197	0.100258
$191$	−13.0902	−0.947171	−0.473586	−	0.880748i	$-0.657041\pi$
$191$	−13.0902	−0.947171	−0.473586	+	0.880748i	$0.657041\pi$
$192$	16.0689	1.15967
$193$	13.2361	0.952753	0.476377	−	0.879241i	$-0.341950\pi$
$193$	13.2361	0.952753	0.476377	+	0.879241i	$0.341950\pi$
$194$	22.7639	1.63436
$195$	−4.94427	−0.354067
$196$	−0.437694	−0.0312639
$197$	−12.4721	−0.888603	−0.444301	−	0.895877i	$-0.646548\pi$
$197$	−12.4721	−0.888603	−0.444301	+	0.895877i	$0.646548\pi$
$198$	0	0
$199$	−23.5623	−1.67029	−0.835144	−	0.550032i	$-0.814616\pi$
$199$	−23.5623	−1.67029	−0.835144	+	0.550032i	$0.814616\pi$
$200$	−10.3262	−0.730175
$201$	−0.583592	−0.0411634
$202$	−19.2705	−1.35587
$203$	1.23607	0.0867550
$204$	−30.0000	−2.10042
$205$	−1.05573	−0.0737352
$206$	5.12461	0.357049
$207$	−2.38197	−0.165558
$208$	−6.47214	−0.448762
$209$	0	0
$210$	4.47214	0.308607
$211$	−9.23607	−0.635837	−0.317919	−	0.948118i	$-0.602984\pi$
$211$	−9.23607	−0.635837	−0.317919	+	0.948118i	$0.602984\pi$
$212$	31.4164	2.15769
$213$	2.11146	0.144675
$214$	−1.70820	−0.116770
$215$	5.00000	0.340997
$216$	12.3607	0.841038
$217$	−20.9443	−1.42179
$218$	−15.5279	−1.05168
$219$	−15.4164	−1.04174
$220$	0	0
$221$	52.3607	3.52216
$222$	5.52786	0.371006
$223$	−8.94427	−0.598953	−0.299476	−	0.954104i	$-0.596812\pi$
$223$	−8.94427	−0.598953	−0.299476	+	0.954104i	$0.596812\pi$
$224$	17.5623	1.17343
$225$	6.79837	0.453225
$226$	−15.1246	−1.00607
$227$	−17.1246	−1.13660	−0.568300	−	0.822821i	$-0.692399\pi$
$227$	−17.1246	−1.13660	−0.568300	+	0.822821i	$0.692399\pi$
$228$	−3.70820	−0.245582
$229$	0.0901699	0.00595860	0.00297930	−	0.999996i	$-0.499052\pi$
$229$	0.0901699	0.00595860	0.00297930	+	0.999996i	$0.499052\pi$
$230$	2.23607	0.147442
$231$	0	0
$232$	−1.05573	−0.0693119
$233$	−3.38197	−0.221560	−0.110780	−	0.993845i	$-0.535335\pi$
$233$	−3.38197	−0.221560	−0.110780	+	0.993845i	$0.535335\pi$
$234$	−21.3050	−1.39275
$235$	3.94427	0.257296
$236$	10.5836	0.688933
$237$	18.4721	1.19989
$238$	−47.3607	−3.06994
$239$	−21.6180	−1.39835	−0.699177	−	0.714948i	$-0.746450\pi$
$239$	−21.6180	−1.39835	−0.699177	+	0.714948i	$0.746450\pi$
$240$	0.763932	0.0493116
$241$	15.5279	1.00024	0.500119	−	0.865957i	$-0.333290\pi$
$241$	15.5279	1.00024	0.500119	+	0.865957i	$0.333290\pi$
$242$	0	0
$243$	−13.5967	−0.872232
$244$	19.1459	1.22569
$245$	−0.0901699	−0.00576075
$246$	4.72136	0.301023
$247$	6.47214	0.411812
$248$	17.8885	1.13592
$249$	12.7639	0.808881
$250$	−13.2918	−0.840647
$251$	7.90983	0.499264	0.249632	−	0.968341i	$-0.419690\pi$
$251$	7.90983	0.499264	0.249632	+	0.968341i	$0.419690\pi$
$252$	11.5623	0.728357
$253$	0	0
$254$	4.47214	0.280607
$255$	−6.18034	−0.387028
$256$	−9.00000	−0.562500
$257$	12.9443	0.807441	0.403721	−	0.914882i	$-0.367717\pi$
$257$	12.9443	0.807441	0.403721	+	0.914882i	$0.367717\pi$
$258$	−22.3607	−1.39212
$259$	5.23607	0.325353
$260$	12.0000	0.744208
$261$	0.695048	0.0430224
$262$	−40.3262	−2.49136
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	6.47214	0.397580
$266$	−5.85410	−0.358938
$267$	−1.52786	−0.0935038
$268$	1.41641	0.0865209
$269$	18.9443	1.15505	0.577526	−	0.816372i	$-0.304018\pi$
$269$	18.9443	1.15505	0.577526	+	0.816372i	$0.304018\pi$
$270$	7.63932	0.464914
$271$	−13.0344	−0.791786	−0.395893	−	0.918297i	$-0.629565\pi$
$271$	−13.0344	−0.791786	−0.395893	+	0.918297i	$0.629565\pi$
$272$	−8.09017	−0.490539
$273$	20.9443	1.26761
$274$	4.14590	0.250463
$275$	0	0
$276$	−6.00000	−0.361158
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	−44.9230	−2.69430
$279$	−11.7771	−0.705076
$280$	−3.61803	−0.216219
$281$	−4.47214	−0.266785	−0.133393	−	0.991063i	$-0.542587\pi$
$281$	−4.47214	−0.266785	−0.133393	+	0.991063i	$0.542587\pi$
$282$	−17.6393	−1.05041
$283$	−17.3820	−1.03325	−0.516625	−	0.856212i	$-0.672812\pi$
$283$	−17.3820	−1.03325	−0.516625	+	0.856212i	$0.672812\pi$
$284$	−5.12461	−0.304090
$285$	−0.763932	−0.0452514
$286$	0	0
$287$	4.47214	0.263982
$288$	9.87539	0.581913
$289$	48.4508	2.85005
$290$	−0.652476	−0.0383147
$291$	−12.5836	−0.737663
$292$	37.4164	2.18963
$293$	−7.52786	−0.439783	−0.219891	−	0.975524i	$-0.570570\pi$
$293$	−7.52786	−0.439783	−0.219891	+	0.975524i	$0.570570\pi$
$294$	0.403252	0.0235181
$295$	2.18034	0.126944
$296$	−4.47214	−0.259938
$297$	0	0
$298$	38.9443	2.25598
$299$	10.4721	0.605619
$300$	17.1246	0.988690
$301$	−21.1803	−1.22081
$302$	6.18034	0.355639
$303$	10.6525	0.611969
$304$	−1.00000	−0.0573539
$305$	3.94427	0.225848
$306$	−26.6312	−1.52240
$307$	9.88854	0.564369	0.282185	−	0.959360i	$-0.408941\pi$
$307$	9.88854	0.564369	0.282185	+	0.959360i	$0.408941\pi$
$308$	0	0
$309$	−2.83282	−0.161153
$310$	11.0557	0.627923
$311$	18.5623	1.05257	0.526286	−	0.850308i	$-0.323584\pi$
$311$	18.5623	1.05257	0.526286	+	0.850308i	$0.323584\pi$
$312$	−17.8885	−1.01274
$313$	−9.61803	−0.543643	−0.271822	−	0.962348i	$-0.587626\pi$
$313$	−9.61803	−0.543643	−0.271822	+	0.962348i	$0.587626\pi$
$314$	−12.4377	−0.701900
$315$	2.38197	0.134209
$316$	−44.8328	−2.52204
$317$	−19.5967	−1.10066	−0.550332	−	0.834946i	$-0.685499\pi$
$317$	−19.5967	−1.10066	−0.550332	+	0.834946i	$0.685499\pi$
$318$	−28.9443	−1.62311
$319$	0	0
$320$	−8.03444	−0.449139
$321$	0.944272	0.0527041
$322$	−9.47214	−0.527861
$323$	8.09017	0.450149
$324$	−7.24922	−0.402735
$325$	−29.8885	−1.65792
$326$	14.1459	0.783469
$327$	8.58359	0.474674
$328$	−3.81966	−0.210905
$329$	−16.7082	−0.921153
$330$	0	0
$331$	−23.7082	−1.30312	−0.651560	−	0.758597i	$-0.725885\pi$
$331$	−23.7082	−1.30312	−0.651560	+	0.758597i	$0.725885\pi$
$332$	−30.9787	−1.70018
$333$	2.94427	0.161345
$334$	−4.06888	−0.222639
$335$	0.291796	0.0159425
$336$	−3.23607	−0.176542
$337$	19.4164	1.05768	0.528840	−	0.848722i	$-0.322627\pi$
$337$	19.4164	1.05768	0.528840	+	0.848722i	$0.322627\pi$
$338$	64.5967	3.51360
$339$	8.36068	0.454090
$340$	15.0000	0.813489
$341$	0	0
$342$	−3.29180	−0.178000
$343$	18.7082	1.01015
$344$	18.0902	0.975357
$345$	−1.23607	−0.0665477
$346$	41.3050	2.22057
$347$	−8.85410	−0.475313	−0.237657	−	0.971349i	$-0.576379\pi$
$347$	−8.85410	−0.475313	−0.237657	+	0.971349i	$0.576379\pi$
$348$	1.75078	0.0938515
$349$	−2.14590	−0.114867	−0.0574336	−	0.998349i	$-0.518292\pi$
$349$	−2.14590	−0.114867	−0.0574336	+	0.998349i	$0.518292\pi$
$350$	27.0344	1.44505
$351$	35.7771	1.90964
$352$	0	0
$353$	0.909830	0.0484254	0.0242127	−	0.999707i	$-0.492292\pi$
$353$	0.909830	0.0484254	0.0242127	+	0.999707i	$0.492292\pi$
$354$	−9.75078	−0.518248
$355$	−1.05573	−0.0560322
$356$	3.70820	0.196534
$357$	26.1803	1.38561
$358$	36.1803	1.91219
$359$	2.20163	0.116197	0.0580987	−	0.998311i	$-0.481496\pi$
$359$	2.20163	0.116197	0.0580987	+	0.998311i	$0.481496\pi$
$360$	−2.03444	−0.107225
$361$	1.00000	0.0526316
$362$	5.52786	0.290538
$363$	0	0
$364$	−50.8328	−2.66436
$365$	7.70820	0.403466
$366$	−17.6393	−0.922022
$367$	18.3262	0.956622	0.478311	−	0.878191i	$-0.341249\pi$
$367$	18.3262	0.956622	0.478311	+	0.878191i	$0.341249\pi$
$368$	−1.61803	−0.0843459
$369$	2.51471	0.130910
$370$	−2.76393	−0.143690
$371$	−27.4164	−1.42339
$372$	−29.6656	−1.53809
$373$	−23.2361	−1.20312	−0.601559	−	0.798828i	$-0.705454\pi$
$373$	−23.2361	−1.20312	−0.601559	+	0.798828i	$0.705454\pi$
$374$	0	0
$375$	7.34752	0.379425
$376$	14.2705	0.735945
$377$	−3.05573	−0.157378
$378$	−32.3607	−1.66445
$379$	−10.6525	−0.547181	−0.273590	−	0.961846i	$-0.588211\pi$
$379$	−10.6525	−0.547181	−0.273590	+	0.961846i	$0.588211\pi$
$380$	1.85410	0.0951134
$381$	−2.47214	−0.126651
$382$	−29.2705	−1.49761
$383$	−7.12461	−0.364051	−0.182025	−	0.983294i	$-0.558265\pi$
$383$	−7.12461	−0.364051	−0.182025	+	0.983294i	$0.558265\pi$
$384$	19.3475	0.987324
$385$	0	0
$386$	29.5967	1.50643
$387$	−11.9098	−0.605411
$388$	30.5410	1.55049
$389$	1.79837	0.0911812	0.0455906	−	0.998960i	$-0.485483\pi$
$389$	1.79837	0.0911812	0.0455906	+	0.998960i	$0.485483\pi$
$390$	−11.0557	−0.559829
$391$	13.0902	0.661998
$392$	−0.326238	−0.0164775
$393$	22.2918	1.12447
$394$	−27.8885	−1.40500
$395$	−9.23607	−0.464717
$396$	0	0
$397$	6.56231	0.329353	0.164676	−	0.986348i	$-0.447342\pi$
$397$	6.56231	0.329353	0.164676	+	0.986348i	$0.447342\pi$
$398$	−52.6869	−2.64096
$399$	3.23607	0.162006
$400$	4.61803	0.230902
$401$	11.7082	0.584680	0.292340	−	0.956314i	$-0.405566\pi$
$401$	11.7082	0.584680	0.292340	+	0.956314i	$0.405566\pi$
$402$	−1.30495	−0.0650851
$403$	51.7771	2.57920
$404$	−25.8541	−1.28629
$405$	−1.49342	−0.0742087
$406$	2.76393	0.137172
$407$	0	0
$408$	−22.3607	−1.10702
$409$	17.1246	0.846758	0.423379	−	0.905953i	$-0.360844\pi$
$409$	17.1246	0.846758	0.423379	+	0.905953i	$0.360844\pi$
$410$	−2.36068	−0.116586
$411$	−2.29180	−0.113046
$412$	6.87539	0.338726
$413$	−9.23607	−0.454477
$414$	−5.32624	−0.261770
$415$	−6.38197	−0.313278
$416$	−43.4164	−2.12866
$417$	24.8328	1.21607
$418$	0	0
$419$	−7.14590	−0.349100	−0.174550	−	0.984648i	$-0.555847\pi$
$419$	−7.14590	−0.349100	−0.174550	+	0.984648i	$0.555847\pi$
$420$	6.00000	0.292770
$421$	13.2361	0.645086	0.322543	−	0.946555i	$-0.395462\pi$
$421$	13.2361	0.645086	0.322543	+	0.946555i	$0.395462\pi$
$422$	−20.6525	−1.00535
$423$	−9.39512	−0.456806
$424$	23.4164	1.13720
$425$	−37.3607	−1.81226
$426$	4.72136	0.228751
$427$	−16.7082	−0.808567
$428$	−2.29180	−0.110778
$429$	0	0
$430$	11.1803	0.539164
$431$	−17.0557	−0.821546	−0.410773	−	0.911738i	$-0.634741\pi$
$431$	−17.0557	−0.821546	−0.410773	+	0.911738i	$0.634741\pi$
$432$	−5.52786	−0.265959
$433$	27.1246	1.30353	0.651763	−	0.758423i	$-0.274030\pi$
$433$	27.1246	1.30353	0.651763	+	0.758423i	$0.274030\pi$
$434$	−46.8328	−2.24805
$435$	0.360680	0.0172933
$436$	−20.8328	−0.997711
$437$	1.61803	0.0774011
$438$	−34.4721	−1.64714
$439$	−8.76393	−0.418280	−0.209140	−	0.977886i	$-0.567066\pi$
$439$	−8.76393	−0.418280	−0.209140	+	0.977886i	$0.567066\pi$
$440$	0	0
$441$	0.214782	0.0102277
$442$	117.082	5.56902
$443$	3.20163	0.152114	0.0760569	−	0.997103i	$-0.475767\pi$
$443$	3.20163	0.152114	0.0760569	+	0.997103i	$0.475767\pi$
$444$	7.41641	0.351967
$445$	0.763932	0.0362139
$446$	−20.0000	−0.947027
$447$	−21.5279	−1.01823
$448$	34.0344	1.60798
$449$	23.5279	1.11035	0.555174	−	0.831734i	$-0.312652\pi$
$449$	23.5279	1.11035	0.555174	+	0.831734i	$0.312652\pi$
$450$	15.2016	0.716612
$451$	0	0
$452$	−20.2918	−0.954446
$453$	−3.41641	−0.160517
$454$	−38.2918	−1.79712
$455$	−10.4721	−0.490941
$456$	−2.76393	−0.129433
$457$	−31.3820	−1.46799	−0.733993	−	0.679157i	$-0.762346\pi$
$457$	−31.3820	−1.46799	−0.733993	+	0.679157i	$0.762346\pi$
$458$	0.201626	0.00942137
$459$	44.7214	2.08741
$460$	3.00000	0.139876
$461$	−34.5066	−1.60713	−0.803566	−	0.595216i	$-0.797067\pi$
$461$	−34.5066	−1.60713	−0.803566	+	0.595216i	$0.797067\pi$
$462$	0	0
$463$	5.38197	0.250121	0.125061	−	0.992149i	$-0.460087\pi$
$463$	5.38197	0.250121	0.125061	+	0.992149i	$0.460087\pi$
$464$	0.472136	0.0219184
$465$	−6.11146	−0.283412
$466$	−7.56231	−0.350317
$467$	9.96556	0.461151	0.230576	−	0.973054i	$-0.425939\pi$
$467$	9.96556	0.461151	0.230576	+	0.973054i	$0.425939\pi$
$468$	−28.5836	−1.32128
$469$	−1.23607	−0.0570763
$470$	8.81966	0.406821
$471$	6.87539	0.316801
$472$	7.88854	0.363100
$473$	0	0
$474$	41.3050	1.89720
$475$	−4.61803	−0.211890
$476$	−63.5410	−2.91240
$477$	−15.4164	−0.705869
$478$	−48.3394	−2.21099
$479$	3.14590	0.143740	0.0718699	−	0.997414i	$-0.477103\pi$
$479$	3.14590	0.143740	0.0718699	+	0.997414i	$0.477103\pi$
$480$	5.12461	0.233905
$481$	−12.9443	−0.590208
$482$	34.7214	1.58151
$483$	5.23607	0.238249
$484$	0	0
$485$	6.29180	0.285696
$486$	−30.4033	−1.37912
$487$	−23.3050	−1.05605	−0.528024	−	0.849229i	$-0.677067\pi$
$487$	−23.3050	−1.05605	−0.528024	+	0.849229i	$0.677067\pi$
$488$	14.2705	0.645996
$489$	−7.81966	−0.353617
$490$	−0.201626	−0.00910854
$491$	−15.8541	−0.715486	−0.357743	−	0.933820i	$-0.616454\pi$
$491$	−15.8541	−0.715486	−0.357743	+	0.933820i	$0.616454\pi$
$492$	6.33437	0.285575
$493$	−3.81966	−0.172029
$494$	14.4721	0.651132
$495$	0	0
$496$	−8.00000	−0.359211
$497$	4.47214	0.200603
$498$	28.5410	1.27895
$499$	−3.56231	−0.159471	−0.0797354	−	0.996816i	$-0.525408\pi$
$499$	−3.56231	−0.159471	−0.0797354	+	0.996816i	$0.525408\pi$
$500$	−17.8328	−0.797508
$501$	2.24922	0.100488
$502$	17.6869	0.789406
$503$	28.0000	1.24846	0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000	1.24846	0.624229	+	0.781241i	$0.285413\pi$
$504$	8.61803	0.383878
$505$	−5.32624	−0.237014
$506$	0	0
$507$	−35.7082	−1.58586
$508$	6.00000	0.266207
$509$	−7.41641	−0.328726	−0.164363	−	0.986400i	$-0.552557\pi$
$509$	−7.41641	−0.328726	−0.164363	+	0.986400i	$0.552557\pi$
$510$	−13.8197	−0.611945
$511$	−32.6525	−1.44446
$512$	11.1803	0.494106
$513$	5.52786	0.244061
$514$	28.9443	1.27668
$515$	1.41641	0.0624144
$516$	−30.0000	−1.32068
$517$	0	0
$518$	11.7082	0.514429
$519$	−22.8328	−1.00225
$520$	8.94427	0.392232
$521$	−23.8885	−1.04658	−0.523288	−	0.852156i	$-0.675295\pi$
$521$	−23.8885	−1.04658	−0.523288	+	0.852156i	$0.675295\pi$
$522$	1.55418	0.0680244
$523$	32.9443	1.44055	0.720276	−	0.693687i	$-0.244015\pi$
$523$	32.9443	1.44055	0.720276	+	0.693687i	$0.244015\pi$
$524$	−54.1033	−2.36351
$525$	−14.9443	−0.652221
$526$	0	0
$527$	64.7214	2.81931
$528$	0	0
$529$	−20.3820	−0.886172
$530$	14.4721	0.628629
$531$	−5.19350	−0.225379
$532$	−7.85410	−0.340519
$533$	−11.0557	−0.478877
$534$	−3.41641	−0.147842
$535$	−0.472136	−0.0204122
$536$	1.05573	0.0456005
$537$	−20.0000	−0.863064
$538$	42.3607	1.82630
$539$	0	0
$540$	10.2492	0.441056
$541$	−22.5623	−0.970029	−0.485015	−	0.874506i	$-0.661186\pi$
$541$	−22.5623	−0.970029	−0.485015	+	0.874506i	$0.661186\pi$
$542$	−29.1459	−1.25192
$543$	−3.05573	−0.131134
$544$	−54.2705	−2.32683
$545$	−4.29180	−0.183840
$546$	46.8328	2.00426
$547$	−35.4164	−1.51430	−0.757148	−	0.653243i	$-0.773408\pi$
$547$	−35.4164	−1.51430	−0.757148	+	0.653243i	$0.773408\pi$
$548$	5.56231	0.237610
$549$	−9.39512	−0.400974
$550$	0	0
$551$	−0.472136	−0.0201137
$552$	−4.47214	−0.190347
$553$	39.1246	1.66375
$554$	13.4164	0.570009
$555$	1.52786	0.0648542
$556$	−60.2705	−2.55604
$557$	−19.6738	−0.833604	−0.416802	−	0.908997i	$-0.636849\pi$
$557$	−19.6738	−0.833604	−0.416802	+	0.908997i	$0.636849\pi$
$558$	−26.3344	−1.11482
$559$	52.3607	2.21462
$560$	1.61803	0.0683744
$561$	0	0
$562$	−10.0000	−0.421825
$563$	30.1803	1.27195	0.635975	−	0.771710i	$-0.280598\pi$
$563$	30.1803	1.27195	0.635975	+	0.771710i	$0.280598\pi$
$564$	−23.6656	−0.996503
$565$	−4.18034	−0.175868
$566$	−38.8673	−1.63371
$567$	6.32624	0.265677
$568$	−3.81966	−0.160269
$569$	−32.2918	−1.35374	−0.676871	−	0.736101i	$-0.736665\pi$
$569$	−32.2918	−1.35374	−0.676871	+	0.736101i	$0.736665\pi$
$570$	−1.70820	−0.0715488
$571$	−5.67376	−0.237440	−0.118720	−	0.992928i	$-0.537879\pi$
$571$	−5.67376	−0.237440	−0.118720	+	0.992928i	$0.537879\pi$
$572$	0	0
$573$	16.1803	0.675943
$574$	10.0000	0.417392
$575$	−7.47214	−0.311610
$576$	19.1378	0.797407
$577$	−34.3607	−1.43045	−0.715227	−	0.698892i	$-0.753677\pi$
$577$	−34.3607	−1.43045	−0.715227	+	0.698892i	$0.753677\pi$
$578$	108.339	4.50632
$579$	−16.3607	−0.679927
$580$	−0.875388	−0.0363485
$581$	27.0344	1.12158
$582$	−28.1378	−1.16635
$583$	0	0
$584$	27.8885	1.15404
$585$	−5.88854	−0.243461
$586$	−16.8328	−0.695357
$587$	−19.4164	−0.801401	−0.400700	−	0.916209i	$-0.631233\pi$
$587$	−19.4164	−0.801401	−0.400700	+	0.916209i	$0.631233\pi$
$588$	0.541020	0.0223113
$589$	8.00000	0.329634
$590$	4.87539	0.200717
$591$	15.4164	0.634146
$592$	2.00000	0.0821995
$593$	24.2705	0.996670	0.498335	−	0.866984i	$-0.333945\pi$
$593$	24.2705	0.996670	0.498335	+	0.866984i	$0.333945\pi$
$594$	0	0
$595$	−13.0902	−0.536645
$596$	52.2492	2.14021
$597$	29.1246	1.19199
$598$	23.4164	0.957568
$599$	8.36068	0.341608	0.170804	−	0.985305i	$-0.445363\pi$
$599$	8.36068	0.341608	0.170804	+	0.985305i	$0.445363\pi$
$600$	12.7639	0.521085
$601$	−38.9443	−1.58857	−0.794285	−	0.607545i	$-0.792154\pi$
$601$	−38.9443	−1.58857	−0.794285	+	0.607545i	$0.792154\pi$
$602$	−47.3607	−1.93028
$603$	−0.695048	−0.0283046
$604$	8.29180	0.337388
$605$	0	0
$606$	23.8197	0.967608
$607$	−2.65248	−0.107661	−0.0538303	−	0.998550i	$-0.517143\pi$
$607$	−2.65248	−0.107661	−0.0538303	+	0.998550i	$0.517143\pi$
$608$	−6.70820	−0.272054
$609$	−1.52786	−0.0619122
$610$	8.81966	0.357098
$611$	41.3050	1.67102
$612$	−35.7295	−1.44428
$613$	37.9787	1.53395	0.766973	−	0.641679i	$-0.221762\pi$
$613$	37.9787	1.53395	0.766973	+	0.641679i	$0.221762\pi$
$614$	22.1115	0.892346
$615$	1.30495	0.0526207
$616$	0	0
$617$	−33.4164	−1.34529	−0.672647	−	0.739964i	$-0.734843\pi$
$617$	−33.4164	−1.34529	−0.672647	+	0.739964i	$0.734843\pi$
$618$	−6.33437	−0.254806
$619$	−5.43769	−0.218559	−0.109280	−	0.994011i	$-0.534854\pi$
$619$	−5.43769	−0.218559	−0.109280	+	0.994011i	$0.534854\pi$
$620$	14.8328	0.595700
$621$	8.94427	0.358921
$622$	41.5066	1.66426
$623$	−3.23607	−0.129650
$624$	8.00000	0.320256
$625$	19.4164	0.776656
$626$	−21.5066	−0.859576
$627$	0	0
$628$	−16.6869	−0.665881
$629$	−16.1803	−0.645152
$630$	5.32624	0.212202
$631$	25.3050	1.00737	0.503687	−	0.863886i	$-0.331976\pi$
$631$	25.3050	1.00737	0.503687	+	0.863886i	$0.331976\pi$
$632$	−33.4164	−1.32923
$633$	11.4164	0.453761
$634$	−43.8197	−1.74030
$635$	1.23607	0.0490519
$636$	−38.8328	−1.53982
$637$	−0.944272	−0.0374134
$638$	0	0
$639$	2.51471	0.0994803
$640$	−9.67376	−0.382389
$641$	−2.11146	−0.0833975	−0.0416988	−	0.999130i	$-0.513277\pi$
$641$	−2.11146	−0.0833975	−0.0416988	+	0.999130i	$0.513277\pi$
$642$	2.11146	0.0833325
$643$	45.0344	1.77599	0.887993	−	0.459858i	$-0.152100\pi$
$643$	45.0344	1.77599	0.887993	+	0.459858i	$0.152100\pi$
$644$	−12.7082	−0.500773
$645$	−6.18034	−0.243351
$646$	18.0902	0.711748
$647$	−21.8885	−0.860527	−0.430264	−	0.902703i	$-0.641579\pi$
$647$	−21.8885	−0.860527	−0.430264	+	0.902703i	$0.641579\pi$
$648$	−5.40325	−0.212260
$649$	0	0
$650$	−66.8328	−2.62140
$651$	25.8885	1.01465
$652$	18.9787	0.743264
$653$	14.5623	0.569867	0.284934	−	0.958547i	$-0.408028\pi$
$653$	14.5623	0.569867	0.284934	+	0.958547i	$0.408028\pi$
$654$	19.1935	0.750525
$655$	−11.1459	−0.435506
$656$	1.70820	0.0666942
$657$	−18.3607	−0.716318
$658$	−37.3607	−1.45647
$659$	−6.29180	−0.245094	−0.122547	−	0.992463i	$-0.539106\pi$
$659$	−6.29180	−0.245094	−0.122547	+	0.992463i	$0.539106\pi$
$660$	0	0
$661$	7.12461	0.277115	0.138558	−	0.990354i	$-0.455753\pi$
$661$	7.12461	0.277115	0.138558	+	0.990354i	$0.455753\pi$
$662$	−53.0132	−2.06041
$663$	−64.7214	−2.51357
$664$	−23.0902	−0.896072
$665$	−1.61803	−0.0627447
$666$	6.58359	0.255109
$667$	−0.763932	−0.0295796
$668$	−5.45898	−0.211214
$669$	11.0557	0.427439
$670$	0.652476	0.0252073
$671$	0	0
$672$	−21.7082	−0.837412
$673$	−27.7082	−1.06807	−0.534036	−	0.845461i	$-0.679325\pi$
$673$	−27.7082	−1.06807	−0.534036	+	0.845461i	$0.679325\pi$
$674$	43.4164	1.67234
$675$	−25.5279	−0.982568
$676$	86.6656	3.33329
$677$	−0.583592	−0.0224293	−0.0112146	−	0.999937i	$-0.503570\pi$
$677$	−0.583592	−0.0224293	−0.0112146	+	0.999937i	$0.503570\pi$
$678$	18.6950	0.717979
$679$	−26.6525	−1.02283
$680$	11.1803	0.428746
$681$	21.1672	0.811128
$682$	0	0
$683$	38.8328	1.48590	0.742948	−	0.669349i	$-0.233427\pi$
$683$	38.8328	1.48590	0.742948	+	0.669349i	$0.233427\pi$
$684$	−4.41641	−0.168866
$685$	1.14590	0.0437825
$686$	41.8328	1.59718
$687$	−0.111456	−0.00425232
$688$	−8.09017	−0.308435
$689$	67.7771	2.58210
$690$	−2.76393	−0.105221
$691$	−10.0344	−0.381728	−0.190864	−	0.981616i	$-0.561129\pi$
$691$	−10.0344	−0.381728	−0.190864	+	0.981616i	$0.561129\pi$
$692$	55.4164	2.10662
$693$	0	0
$694$	−19.7984	−0.751536
$695$	−12.4164	−0.470981
$696$	1.30495	0.0494641
$697$	−13.8197	−0.523457
$698$	−4.79837	−0.181621
$699$	4.18034	0.158115
$700$	36.2705	1.37090
$701$	51.5066	1.94538	0.972688	−	0.232115i	$-0.0745646\pi$
$701$	51.5066	1.94538	0.972688	+	0.232115i	$0.0745646\pi$
$702$	80.0000	3.01941
$703$	−2.00000	−0.0754314
$704$	0	0
$705$	−4.87539	−0.183618
$706$	2.03444	0.0765672
$707$	22.5623	0.848543
$708$	−13.0820	−0.491653
$709$	−3.09017	−0.116054	−0.0580269	−	0.998315i	$-0.518481\pi$
$709$	−3.09017	−0.116054	−0.0580269	+	0.998315i	$0.518481\pi$
$710$	−2.36068	−0.0885947
$711$	22.0000	0.825064
$712$	2.76393	0.103583
$713$	12.9443	0.484767
$714$	58.5410	2.19084
$715$	0	0
$716$	48.5410	1.81406
$717$	26.7214	0.997927
$718$	4.92299	0.183724
$719$	26.9787	1.00614	0.503068	−	0.864247i	$-0.332204\pi$
$719$	26.9787	1.00614	0.503068	+	0.864247i	$0.332204\pi$
$720$	0.909830	0.0339074
$721$	−6.00000	−0.223452
$722$	2.23607	0.0832178
$723$	−19.1935	−0.713814
$724$	7.41641	0.275629
$725$	2.18034	0.0809758
$726$	0	0
$727$	−20.8541	−0.773436	−0.386718	−	0.922198i	$-0.626391\pi$
$727$	−20.8541	−0.773436	−0.386718	+	0.922198i	$0.626391\pi$
$728$	−37.8885	−1.40424
$729$	24.0557	0.890953
$730$	17.2361	0.637935
$731$	65.4508	2.42079
$732$	−23.6656	−0.874707
$733$	42.1459	1.55669	0.778347	−	0.627834i	$-0.216058\pi$
$733$	42.1459	1.55669	0.778347	+	0.627834i	$0.216058\pi$
$734$	40.9787	1.51255
$735$	0.111456	0.00411112
$736$	−10.8541	−0.400088
$737$	0	0
$738$	5.62306	0.206988
$739$	−35.2705	−1.29745	−0.648723	−	0.761024i	$-0.724697\pi$
$739$	−35.2705	−1.29745	−0.648723	+	0.761024i	$0.724697\pi$
$740$	−3.70820	−0.136316
$741$	−8.00000	−0.293887
$742$	−61.3050	−2.25058
$743$	−27.8885	−1.02313	−0.511566	−	0.859244i	$-0.670934\pi$
$743$	−27.8885	−1.02313	−0.511566	+	0.859244i	$0.670934\pi$
$744$	−22.1115	−0.810645
$745$	10.7639	0.394360
$746$	−51.9574	−1.90230
$747$	15.2016	0.556198
$748$	0	0
$749$	2.00000	0.0730784
$750$	16.4296	0.599923
$751$	11.2361	0.410010	0.205005	−	0.978761i	$-0.434279\pi$
$751$	11.2361	0.410010	0.205005	+	0.978761i	$0.434279\pi$
$752$	−6.38197	−0.232726
$753$	−9.77709	−0.356297
$754$	−6.83282	−0.248836
$755$	1.70820	0.0621679
$756$	−43.4164	−1.57904
$757$	6.36068	0.231183	0.115591	−	0.993297i	$-0.463124\pi$
$757$	6.36068	0.231183	0.115591	+	0.993297i	$0.463124\pi$
$758$	−23.8197	−0.865169
$759$	0	0
$760$	1.38197	0.0501292
$761$	51.3050	1.85980	0.929902	−	0.367809i	$-0.119892\pi$
$761$	51.3050	1.85980	0.929902	+	0.367809i	$0.119892\pi$
$762$	−5.52786	−0.200253
$763$	18.1803	0.658172
$764$	−39.2705	−1.42076
$765$	−7.36068	−0.266126
$766$	−15.9311	−0.575615
$767$	22.8328	0.824445
$768$	11.1246	0.401425
$769$	−11.7426	−0.423451	−0.211725	−	0.977329i	$-0.567908\pi$
$769$	−11.7426	−0.423451	−0.211725	+	0.977329i	$0.567908\pi$
$770$	0	0
$771$	−16.0000	−0.576226
$772$	39.7082	1.42913
$773$	−48.7639	−1.75392	−0.876958	−	0.480566i	$-0.840431\pi$
$773$	−48.7639	−1.75392	−0.876958	+	0.480566i	$0.840431\pi$
$774$	−26.6312	−0.957238
$775$	−36.9443	−1.32708
$776$	22.7639	0.817178
$777$	−6.47214	−0.232187
$778$	4.02129	0.144170
$779$	−1.70820	−0.0612028
$780$	−14.8328	−0.531100
$781$	0	0
$782$	29.2705	1.04671
$783$	−2.60990	−0.0932703
$784$	0.145898	0.00521064
$785$	−3.43769	−0.122697
$786$	49.8460	1.77795
$787$	15.5279	0.553509	0.276754	−	0.960941i	$-0.410741\pi$
$787$	15.5279	0.553509	0.276754	+	0.960941i	$0.410741\pi$
$788$	−37.4164	−1.33290
$789$	0	0
$790$	−20.6525	−0.734782
$791$	17.7082	0.629631
$792$	0	0
$793$	41.3050	1.46678
$794$	14.6738	0.520752
$795$	−8.00000	−0.283731
$796$	−70.6869	−2.50543
$797$	14.9443	0.529353	0.264677	−	0.964337i	$-0.414735\pi$
$797$	14.9443	0.529353	0.264677	+	0.964337i	$0.414735\pi$
$798$	7.23607	0.256154
$799$	51.6312	1.82658
$800$	30.9787	1.09526
$801$	−1.81966	−0.0642945
$802$	26.1803	0.924460
$803$	0	0
$804$	−1.75078	−0.0617451
$805$	−2.61803	−0.0922736
$806$	115.777	4.07807
$807$	−23.4164	−0.824297
$808$	−19.2705	−0.677934
$809$	−16.6869	−0.586681	−0.293340	−	0.956008i	$-0.594767\pi$
$809$	−16.6869	−0.586681	−0.293340	+	0.956008i	$0.594767\pi$
$810$	−3.33939	−0.117334
$811$	40.6525	1.42750	0.713751	−	0.700400i	$-0.246995\pi$
$811$	40.6525	1.42750	0.713751	+	0.700400i	$0.246995\pi$
$812$	3.70820	0.130132
$813$	16.1115	0.565053
$814$	0	0
$815$	3.90983	0.136955
$816$	10.0000	0.350070
$817$	8.09017	0.283039
$818$	38.2918	1.33884
$819$	24.9443	0.871623
$820$	−3.16718	−0.110603
$821$	−39.3262	−1.37250	−0.686248	−	0.727368i	$-0.740743\pi$
$821$	−39.3262	−1.37250	−0.686248	+	0.727368i	$0.740743\pi$
$822$	−5.12461	−0.178741
$823$	−10.0344	−0.349779	−0.174889	−	0.984588i	$-0.555957\pi$
$823$	−10.0344	−0.349779	−0.174889	+	0.984588i	$0.555957\pi$
$824$	5.12461	0.178524
$825$	0	0
$826$	−20.6525	−0.718592
$827$	−36.7214	−1.27693	−0.638463	−	0.769652i	$-0.720430\pi$
$827$	−36.7214	−1.27693	−0.638463	+	0.769652i	$0.720430\pi$
$828$	−7.14590	−0.248337
$829$	15.3475	0.533042	0.266521	−	0.963829i	$-0.414126\pi$
$829$	15.3475	0.533042	0.266521	+	0.963829i	$0.414126\pi$
$830$	−14.2705	−0.495337
$831$	−7.41641	−0.257272
$832$	−84.1378	−2.91695
$833$	−1.18034	−0.0408964
$834$	55.5279	1.92277
$835$	−1.12461	−0.0389188
$836$	0	0
$837$	44.2229	1.52857
$838$	−15.9787	−0.551976
$839$	9.70820	0.335164	0.167582	−	0.985858i	$-0.446404\pi$
$839$	9.70820	0.335164	0.167582	+	0.985858i	$0.446404\pi$
$840$	4.47214	0.154303
$841$	−28.7771	−0.992313
$842$	29.5967	1.01997
$843$	5.52786	0.190390
$844$	−27.7082	−0.953756
$845$	17.8541	0.614200
$846$	−21.0081	−0.722274
$847$	0	0
$848$	−10.4721	−0.359615
$849$	21.4853	0.737373
$850$	−83.5410	−2.86543
$851$	−3.23607	−0.110931
$852$	6.33437	0.217012
$853$	−2.20163	−0.0753822	−0.0376911	−	0.999289i	$-0.512000\pi$
$853$	−2.20163	−0.0753822	−0.0376911	+	0.999289i	$0.512000\pi$
$854$	−37.3607	−1.27846
$855$	−0.909830	−0.0311155
$856$	−1.70820	−0.0583852
$857$	17.2361	0.588773	0.294386	−	0.955686i	$-0.404885\pi$
$857$	17.2361	0.588773	0.294386	+	0.955686i	$0.404885\pi$
$858$	0	0
$859$	18.7426	0.639491	0.319745	−	0.947503i	$-0.396403\pi$
$859$	18.7426	0.639491	0.319745	+	0.947503i	$0.396403\pi$
$860$	15.0000	0.511496
$861$	−5.52786	−0.188389
$862$	−38.1378	−1.29898
$863$	12.8328	0.436834	0.218417	−	0.975856i	$-0.429911\pi$
$863$	12.8328	0.436834	0.218417	+	0.975856i	$0.429911\pi$
$864$	−37.0820	−1.26156
$865$	11.4164	0.388170
$866$	60.6525	2.06106
$867$	−59.8885	−2.03392
$868$	−62.8328	−2.13268
$869$	0	0
$870$	0.806504	0.0273431
$871$	3.05573	0.103539
$872$	−15.5279	−0.525840
$873$	−14.9868	−0.507228
$874$	3.61803	0.122382
$875$	15.5623	0.526102
$876$	−46.2492	−1.56262
$877$	−41.8885	−1.41448	−0.707238	−	0.706976i	$-0.750059\pi$
$877$	−41.8885	−1.41448	−0.707238	+	0.706976i	$0.750059\pi$
$878$	−19.5967	−0.661358
$879$	9.30495	0.313848
$880$	0	0
$881$	49.7771	1.67703	0.838516	−	0.544876i	$-0.183423\pi$
$881$	49.7771	1.67703	0.838516	+	0.544876i	$0.183423\pi$
$882$	0.480267	0.0161714
$883$	32.5066	1.09393	0.546967	−	0.837154i	$-0.315782\pi$
$883$	32.5066	1.09393	0.546967	+	0.837154i	$0.315782\pi$
$884$	157.082	5.28324
$885$	−2.69505	−0.0905931
$886$	7.15905	0.240513
$887$	35.3050	1.18542	0.592712	−	0.805414i	$-0.298057\pi$
$887$	35.3050	1.18542	0.592712	+	0.805414i	$0.298057\pi$
$888$	5.52786	0.185503
$889$	−5.23607	−0.175612
$890$	1.70820	0.0572591
$891$	0	0
$892$	−26.8328	−0.898429
$893$	6.38197	0.213564
$894$	−48.1378	−1.60997
$895$	10.0000	0.334263
$896$	40.9787	1.36900
$897$	−12.9443	−0.432197
$898$	52.6099	1.75562
$899$	−3.77709	−0.125973
$900$	20.3951	0.679837
$901$	84.7214	2.82248
$902$	0	0
$903$	26.1803	0.871227
$904$	−15.1246	−0.503037
$905$	1.52786	0.0507879
$906$	−7.63932	−0.253799
$907$	19.3050	0.641010	0.320505	−	0.947247i	$-0.396147\pi$
$907$	19.3050	0.641010	0.320505	+	0.947247i	$0.396147\pi$
$908$	−51.3738	−1.70490
$909$	12.6869	0.420798
$910$	−23.4164	−0.776246
$911$	26.1803	0.867393	0.433697	−	0.901059i	$-0.357209\pi$
$911$	26.1803	0.867393	0.433697	+	0.901059i	$0.357209\pi$
$912$	1.23607	0.0409303
$913$	0	0
$914$	−70.1722	−2.32109
$915$	−4.87539	−0.161175
$916$	0.270510	0.00893790
$917$	47.2148	1.55917
$918$	100.000	3.30049
$919$	52.5066	1.73203	0.866016	−	0.500016i	$-0.166673\pi$
$919$	52.5066	1.73203	0.866016	+	0.500016i	$0.166673\pi$
$920$	2.23607	0.0737210
$921$	−12.2229	−0.402759
$922$	−77.1591	−2.54110
$923$	−11.0557	−0.363904
$924$	0	0
$925$	9.23607	0.303680
$926$	12.0344	0.395476
$927$	−3.37384	−0.110811
$928$	3.16718	0.103968
$929$	40.8541	1.34038	0.670190	−	0.742190i	$-0.266213\pi$
$929$	40.8541	1.34038	0.670190	+	0.742190i	$0.266213\pi$
$930$	−13.6656	−0.448114
$931$	−0.145898	−0.00478161
$932$	−10.1459	−0.332340
$933$	−22.9443	−0.751162
$934$	22.2837	0.729144
$935$	0	0
$936$	−21.3050	−0.696374
$937$	−9.43769	−0.308316	−0.154158	−	0.988046i	$-0.549266\pi$
$937$	−9.43769	−0.308316	−0.154158	+	0.988046i	$0.549266\pi$
$938$	−2.76393	−0.0902456
$939$	11.8885	0.387968
$940$	11.8328	0.385944
$941$	−10.5836	−0.345015	−0.172508	−	0.985008i	$-0.555187\pi$
$941$	−10.5836	−0.345015	−0.172508	+	0.985008i	$0.555187\pi$
$942$	15.3738	0.500907
$943$	−2.76393	−0.0900060
$944$	−3.52786	−0.114822
$945$	−8.94427	−0.290957
$946$	0	0
$947$	−10.6869	−0.347278	−0.173639	−	0.984809i	$-0.555553\pi$
$947$	−10.6869	−0.347278	−0.173639	+	0.984809i	$0.555553\pi$
$948$	55.4164	1.79984
$949$	80.7214	2.62033
$950$	−10.3262	−0.335027
$951$	24.2229	0.785482
$952$	−47.3607	−1.53497
$953$	−27.7082	−0.897557	−0.448778	−	0.893643i	$-0.648141\pi$
$953$	−27.7082	−0.897557	−0.448778	+	0.893643i	$0.648141\pi$
$954$	−34.4721	−1.11608
$955$	−8.09017	−0.261792
$956$	−64.8541	−2.09753
$957$	0	0
$958$	7.03444	0.227273
$959$	−4.85410	−0.156747
$960$	9.93112	0.320525
$961$	33.0000	1.06452
$962$	−28.9443	−0.933201
$963$	1.12461	0.0362401
$964$	46.5836	1.50036
$965$	8.18034	0.263334
$966$	11.7082	0.376705
$967$	−31.1459	−1.00158	−0.500792	−	0.865567i	$-0.666958\pi$
$967$	−31.1459	−1.00158	−0.500792	+	0.865567i	$0.666958\pi$
$968$	0	0
$969$	−10.0000	−0.321246
$970$	14.0689	0.451725
$971$	16.1803	0.519252	0.259626	−	0.965709i	$-0.416401\pi$
$971$	16.1803	0.519252	0.259626	+	0.965709i	$0.416401\pi$
$972$	−40.7902	−1.30835
$973$	52.5967	1.68617
$974$	−52.1115	−1.66976
$975$	36.9443	1.18316
$976$	−6.38197	−0.204282
$977$	1.12461	0.0359795	0.0179898	−	0.999838i	$-0.494273\pi$
$977$	1.12461	0.0359795	0.0179898	+	0.999838i	$0.494273\pi$
$978$	−17.4853	−0.559118
$979$	0	0
$980$	−0.270510	−0.00864112
$981$	10.2229	0.326392
$982$	−35.4508	−1.13128
$983$	−23.8197	−0.759729	−0.379865	−	0.925042i	$-0.624029\pi$
$983$	−23.8197	−0.759729	−0.379865	+	0.925042i	$0.624029\pi$
$984$	4.72136	0.150511
$985$	−7.70820	−0.245604
$986$	−8.54102	−0.272001
$987$	20.6525	0.657376
$988$	19.4164	0.617718
$989$	13.0902	0.416243
$990$	0	0
$991$	−22.6525	−0.719580	−0.359790	−	0.933033i	$-0.617152\pi$
$991$	−22.6525	−0.719580	−0.359790	+	0.933033i	$0.617152\pi$
$992$	−53.6656	−1.70389
$993$	29.3050	0.929965
$994$	10.0000	0.317181
$995$	−14.5623	−0.461656
$996$	38.2918	1.21332
$997$	38.2148	1.21027	0.605137	−	0.796121i	$-0.293118\pi$
$997$	38.2148	1.21027	0.605137	+	0.796121i	$0.293118\pi$
$998$	−7.96556	−0.252145
$999$	−11.0557	−0.349788

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2299.2.a.g.1.2		2
11.5	even	5		209.2.f.a.58.1	✓	4
11.9	even	5		209.2.f.a.191.1	yes	4
11.10	odd	2		2299.2.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.f.a.58.1	✓	4	11.5	even	5
209.2.f.a.191.1	yes	4	11.9	even	5
2299.2.a.g.1.2		2	1.1	even	1	trivial
2299.2.a.h.1.1		2	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:	\( N \)	\(=\)	\( 2299 = 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2299.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} -1.23607 q^{3} +3.00000 q^{4} +0.618034 q^{5} -2.76393 q^{6} -2.61803 q^{7} +2.23607 q^{8} -1.47214 q^{9} +1.38197 q^{10} -3.70820 q^{12} +6.47214 q^{13} -5.85410 q^{14} -0.763932 q^{15} -1.00000 q^{16} +8.09017 q^{17} -3.29180 q^{18} +1.00000 q^{19} +1.85410 q^{20} +3.23607 q^{21} +1.61803 q^{23} -2.76393 q^{24} -4.61803 q^{25} +14.4721 q^{26} +5.52786 q^{27} -7.85410 q^{28} -0.472136 q^{29} -1.70820 q^{30} +8.00000 q^{31} -6.70820 q^{32} +18.0902 q^{34} -1.61803 q^{35} -4.41641 q^{36} -2.00000 q^{37} +2.23607 q^{38} -8.00000 q^{39} +1.38197 q^{40} -1.70820 q^{41} +7.23607 q^{42} +8.09017 q^{43} -0.909830 q^{45} +3.61803 q^{46} +6.38197 q^{47} +1.23607 q^{48} -0.145898 q^{49} -10.3262 q^{50} -10.0000 q^{51} +19.4164 q^{52} +10.4721 q^{53} +12.3607 q^{54} -5.85410 q^{56} -1.23607 q^{57} -1.05573 q^{58} +3.52786 q^{59} -2.29180 q^{60} +6.38197 q^{61} +17.8885 q^{62} +3.85410 q^{63} -13.0000 q^{64} +4.00000 q^{65} +0.472136 q^{67} +24.2705 q^{68} -2.00000 q^{69} -3.61803 q^{70} -1.70820 q^{71} -3.29180 q^{72} +12.4721 q^{73} -4.47214 q^{74} +5.70820 q^{75} +3.00000 q^{76} -17.8885 q^{78} -14.9443 q^{79} -0.618034 q^{80} -2.41641 q^{81} -3.81966 q^{82} -10.3262 q^{83} +9.70820 q^{84} +5.00000 q^{85} +18.0902 q^{86} +0.583592 q^{87} +1.23607 q^{89} -2.03444 q^{90} -16.9443 q^{91} +4.85410 q^{92} -9.88854 q^{93} +14.2705 q^{94} +0.618034 q^{95} +8.29180 q^{96} +10.1803 q^{97} -0.326238 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 6 q^{4} - q^{5} - 10 q^{6} - 3 q^{7} + 6 q^{9} + 5 q^{10} + 6 q^{12} + 4 q^{13} - 5 q^{14} - 6 q^{15} - 2 q^{16} + 5 q^{17} - 20 q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{21} + q^{23} - 10 q^{24}+ \cdots + 15 q^{98}+O(q^{100}) \) 2 * q + 2 * q^3 + 6 * q^4 - q^5 - 10 * q^6 - 3 * q^7 + 6 * q^9 + 5 * q^10 + 6 * q^12 + 4 * q^13 - 5 * q^14 - 6 * q^15 - 2 * q^16 + 5 * q^17 - 20 * q^18 + 2 * q^19 - 3 * q^20 + 2 * q^21 + q^23 - 10 * q^24 - 7 * q^25 + 20 * q^26 + 20 * q^27 - 9 * q^28 + 8 * q^29 + 10 * q^30 + 16 * q^31 + 25 * q^34 - q^35 + 18 * q^36 - 4 * q^37 - 16 * q^39 + 5 * q^40 + 10 * q^41 + 10 * q^42 + 5 * q^43 - 13 * q^45 + 5 * q^46 + 15 * q^47 - 2 * q^48 - 7 * q^49 - 5 * q^50 - 20 * q^51 + 12 * q^52 + 12 * q^53 - 20 * q^54 - 5 * q^56 + 2 * q^57 - 20 * q^58 + 16 * q^59 - 18 * q^60 + 15 * q^61 + q^63 - 26 * q^64 + 8 * q^65 - 8 * q^67 + 15 * q^68 - 4 * q^69 - 5 * q^70 + 10 * q^71 - 20 * q^72 + 16 * q^73 - 2 * q^75 + 6 * q^76 - 12 * q^79 + q^80 + 22 * q^81 - 30 * q^82 - 5 * q^83 + 6 * q^84 + 10 * q^85 + 25 * q^86 + 28 * q^87 - 2 * q^89 + 25 * q^90 - 16 * q^91 + 3 * q^92 + 16 * q^93 - 5 * q^94 - q^95 + 30 * q^96 - 2 * q^97 + 15 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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